4. Discrete Probability Distributions

Transcription

1 4. Discrete Probabilit Distributions 4.. Random Variables and Their Probabilit Distributions Most of the exeriments we encounter generate outcomes that can be interreted in terms of real numbers, such as heights of children, numbers of voters favoring various candidates, tensile strength of wires, and numbers of accidents at secified intersections. These numerical outcomes, whose values can change from exeriment to exeriment, are called random variables. We will look at an illustrative examle of a random variable before we attemt a more formal definition. A section of an electrical circuit has two relas, numbered and, oerating in arallel. The current will flow when a switch is thrown if either one or both of the relas close. The robabilit that a rela will close roerl is 0.8, and the robabilit is the same for each rela. The relas oerate indeendentl, we assume. Let E i denote the event that rela i closes roerl when the switch is thrown. Then P(E i ) 0.8. When the switch is thrown, a numerical outcome of some interest to the oerator of this sstem is, the number of relas that close roerl. Now, can take on onl three ossible values, because the number of relas that close must be 0,, or. We can find the robabilities associated with these values of b relating them to the underling events E i. Thus, we have P( 0) P( EE ) P( E) P( E). 0.(0.) 0.04 because 0 means that neither rela closes and the relas oerate indeendentl. Similarl, P( ) P( E E E E ) and P( E E P( E ) P( E 0.8(0.) 0,(0.8) 0.3 P( ) P( E E ) P( E E 0.8(0.8) ) ) P( E ) P( E P( E ) P( E 0.64 The values of, along with their robabilities, are more useful for keeing track of the oeration of this sstem than are the underling events E i, because the number of roerl closing relas is the ke to whether the sstem will work. The current will flow if is equal to at least, and this event has robabilit ) ) )

2 P( ) P( P( ) P( ) 0.96 Notice that we have maed the outcomes of an exeriment into a set of three meaningful real numbers and have attached a robabilit to each. Such situations rovide the motivation for Definitions 4. and 4.. or ) Definition 4.. A random variable is a real-valued function whose domain is a samle sace. Random variables will be denoted b uer-case letters, usuall toward the end of the alhabet, such as, Y, and Z. The actual values that random variables can assume will be denoted b lower-case letters, such as x,, and z. We can then talk about the robabilit that takes on the value x, P( x), which is denoted b (x). In the rela examle, the random variable has onl three ossible values, and it is a relativel simle matter to assign robabilities to these values. Such a random variable is called discrete. Definition 4.. A random variable is said to be discrete if it can take on onl a finite number or a countabl infinite number of ossible values x. The robabilit function of, denoted b (x), assigns robabilit to each value x of so that following conditions are satisfied:. P ( x) ( x) 0.. P ( x), where the sum is over all ossible values of x. x The robabilit function is sometimes called the robabilit mass function of, to denote the idea that a mass of robabilit is associated with values for discrete oints. It is often convenient to list the robabilities for a discrete random variable in a table. With defined as the number of closed relas in the roblem just discussed, the table is as follows: x (x) Total.00 This listing constitutes one wa of reresenting the robabilit distribution of. Notice that the robabilit function (x) satisfies two roerties:. 0 ( x) for an x.

3 3. ( x), where the sum is over all ossible values of x. x In general, a function is a robabilit function if and onl if the above two conditions are satisfied. Bar grahs are used to disla the robabilit functions for discrete random variables. The robabilit distribution of the number of closed relas discussed above is shown in Figure 4.. Figure 4.. Grah of a robabilit mass function Functional forms for some robabilit functions that have been useful for modeling real-life data will be given in later sections. We now illustrate another method for arriving at a tabular resentation of a discrete robabilit distribution. Examle 4.: A local video store eriodicall uts its used movies in a bin and offers to sell them to customers at a reduced rice. Twelve coies of a oular movie have just been added to the bin, but three of these are defective. A customer randoml selects two of the coies for gifts. Let be the number of defective movies the customer urchased. Find the robabilit function for and grah the function. Solution: The exeriment consists of two selections, each of which can result in one of two outcomes. Let D i denote the event that the ith movie selected is defective; thus, D i denotes the event that it is good. The robabilit of selecting two good movies ( 0) is

4 4 P ( D ) D P( D on st) P( D on nd D on st) The multilicative law of robabilit is used, and the robabilit for the second selection deends on what haened on the first selection. Other ossibilities for outcomes will result in other values of. These outcomes are convenientl listed on the tree in Figure 3.. The robabilities for the various selections are given on the branches of the tree. Figure 4.. Figure 4.. Outcomes for Examle 4. Clearl, has three ossible outcomes, with robabilities as follows: x (x) Total.00 The robabilities are grahed in the figure below.

5 5 Tr to envision this concet extended to more selections from bins of various structures. We sometimes stud the behavior of random variables b looking at the cumulative robabilities; that is, for an random variable, we ma look at P( b) for an real number b. This is, the cumulative robabilit for evaluated at b. Thus, we can define a function F(b) as F(b) P( b). Definition 4.3. The distribution function F(b) for a random variable is If is discrete, F(b) P( b) F ( b) ( x) b x where (x) is the robabilit function. The distribution function is often called the cumulative distribution function (c.d.f). The random variable, denoting the number of relas that close roerl (as defined at the beginning of this section), has a robabilit distribution given b P( 0) 0.04 P( ) 0.3 P( ) 0.64

6 6 Because ositive robabilit is associated onl for x 0,, or, the distribution function changes values onl at those oints. For values of b at least, but less than, the P( b) P( ). For examle, we can see that P(.5) P(.9) P( ) 0.36 The distribution function for this random variable then has the form 0, x < , 0 x < F ( x) 0.36, x <, x The function is grahed in Figure 4.3. Figure 4.3. Distribution Function Notice that the distribution function is a ste function and is defined for all real numbers. This is true for all discrete random variables. The distribution function is discontinuous at oints of ositive robabilit. Because the outcomes 0,, and have ositive robabilit associated with them, the distribution function is discontinuous at those oints. The change in the value of the function at a oint (the height of the ste) is the robabilit associated with that value x. Since the outcome of is the most robable (() 0.64), the height of the ste at this oint is the largest. Although the function has oints of discontinuit, it is right-hand continuous at all oints. To see this, consider. As we aroach from the left, we have lim F( h) 0.36 F() ; that is, the distribution h 0 function F is right-hand continuous. However, if we aroach from the left, we find lim F( h) F(), giving us that F is not left-hand continuous. Because h 0 a function must be both left- and right-hand continuous to be continuous, F is not continuous at. In general, a distribution function is defined for the whole real line. Ever distribution function must satisf four roerties; similarl, an function satisfing the following four roerties is a distribution function.

7 7. lim F( x) 0 x. lim F( x) x 3. The distribution function is a non-decreasing function; that is, if a < b, F(a) F(b). The distribution function can remain constant, but it cannot decrease, as we increase from a to b. 4. The distribution function is right-hand continuous; that is, lim F( x h) F( x) We have alread seen that, given a robabilit mass function, we can determine the distribution function. For an distribution function, we can also determine the robabilit function. h 0 Examle 4.: A large universit uses some of the student fees to offer free use of its Health Center to all students. Let be the number of times that a randoml selected student visits the Health Center during a semester. Based on historical data, the distribution function of is given below. 0, x < 0 0.6, 0 x < F ( x) 0.8, x < 0.95, x < 3, x 3 For the function above,. Grah F.. Verif that F is a distribution function. 3. Find the robabilit function associated with F. Solution:.

8 8. To verif F is a distribution function, we must confirm that the function satisfies the four conditions of a distribution function. () Because F is zero for all values x less than 0, lim F( x) 0. x () Similarl F is one for all values of x that are 3 or greater; therefore, lim F( x). x (3) The function F is non-decreasing. There are man oints for which it is not increasing, but as x increases, F(x) either remains constant or increases. (4) The function is discontinuous at four oints: 0,,, and 3. At each of these oints, F is right-hand continuous. As an examle, for, lim F( h) 0.95 F(). h 0 Because F satisfies the four conditions, it is a distribution function. 3. The oints of ositive robabilit occur at the oints of discontinuit: 0,,, and 3. Further, the robabilit is the height of the jum at that oint. This gives us the following robabilities. x (x) Exercises 4.. Circuit boards from two assembl lines set u to roduce identical boards are mixed in one storage tra. As insectors examine the boards, the find that it is difficult to determine whether a board comes from line A or line B. A robabilistic assessment of this question is often helful. Suose that the storage tra contains ten circuit boards, of which six came from line A and four from line B. An insector selects two of these identical-looking boards for insection. He is interested in, the number of insected boards from line A. a. Find the robabilit function for. b. Grah the robabilit function of. c. Find the distribution function of. d. Grah the distribution function of. 4.. Among twelve alicants for an oen osition, seven are women and five are men. Suose that three alicants are randoml selected from the alicant ool for final interviews. Let be the number of female alicants among the final three. a. Find the robabilit function for. b. Grah the robabilit function of. c. Find the distribution function of. d. Grah the distribution function of.

9 The median annual income for heads of households in a certain cit is $44,000. Four such heads of household are randoml selected for inclusion in an oinion oll. Let be the number (out of the four) who have annual incomes below $44,000. a. Find the robabilit distribution of. b. Grah the robabilit distribution of. c. Find the distribution function of. d. Grah the distribution function of. e. Is it unusual to see all four below $44,000 in this te of oll? (What is the robabilit of this event?) 4.4. At a miniature golf course, laers record the strokes required to make each hole. If the ball is not in the hole after five strokes, the laer is to ick u the ball and record six strokes. The owner is concerned about the flow of laers at hole 7. (She thinks that laers tend to get backed u at that hole.). She has determined that the distribution function of, the number of strokes a laer takes to comlete hole 7 to be 0, x < 0.05, x < 0.5, x < 3 F ( x) 0.35, 3 x < , 4 x < , 5 x < 6, x 6 a. Grah the distribution function of. b. Find the robabilit function of. c. Grah the robabilit function of. d. Based on (a) through (c), are the owner s concerns substantiated? 4.5. In 005, Derrek Lee led the National Baseball League with a batting average, meaning that he got a hit on 33.5% of his official times at bat. In a tical game, he had three official at bats. a. Find the robabilit distribution for, the number of hits Boggs got in a tical game. b. What assumtions are involved in the answer? Are the assumtions reasonable? c. Is it unusual for a good hitter to go 0 for 3 in one game? 4.6. A commercial building has three entrances, numbered I, II, and III. Four eole enter the building at 9:00 a.m. Let denote the number who select entrance I. Assuming that the eole choose entrances indeendentl and at random, find the robabilit distribution for. Were an additional assumtions necessar for our answer?

10 In 00, 33.9% of all fires were structure fires. Of these, 78% of these were residential fires. The causes of structure fire and the numbers of fires during 00 for each cause are dislaed in the table below. Suose that four indeendent structure fires are reorted in one da, and let denote the number, out of the four, that are caused b cooking. Cause of Fire Number of Fires Cooking 9,706 Chimne Fires 8,638 Incinerator 84 Fuel Burner 3,6 Commercial Comactor 46 Trash/Rubbish 9,906 a. Find the robabilit distribution for, in tabular form. b. Find the robabilit that at least one of the four fires was caused b cooking Observers have noticed that the distribution function of, the number of commercial vehicles that cross a certain toll bridge during a minute is as follows: 0, x < 0 0.0, 0 x < F ( x) 0.50, x < 0.85, x < 4, x 4 a. Grah the distribution function of. b. Find the robabilit function of. c. Grah the robabilit function of Of the eole who enter a blood bank to donate blood, in 3 have te O blood, and in 0 have te O - blood. For the next three eole entering the blood bank, let denote the number with O blood, and let Y denote the number with O - blood. Assume the indeendence among the eole with resect to blood te. a. Find the robabilit distribution for and Y. b. Find the robabilit distribution of Y, the number of eole with te O blood Dail sales records for a car dealershi show that it will sell 0,,, or 3 cars, with robabilities as listed: Number of Sales 0 3 Probabilit

11 a. Find the robabilit distribution for, the number of sales in a two-da eriod, assuming the sales are indeendent from da to da. b. Find the robabilit that at least one sale is made in the next two das. 4.. Four microchis are to be laced in a comuter. Two of the four chis are randoml selected for insection before the comuter is assembled. Let denote the number of defective chis found among the two insected. Find the robabilit distribution for for the following events. a. Two of the microchis were defective. b. One of the four microchis was defective. c. None of the microchis was defective. 4.. When turned on, each of the three switches in the accomaning diagram works roerl with robabilit 0.9. If a switch is working roerl, current can flow through it when it is turned on. Find the robabilit distribution for Y, the number of closed aths from a to b, when all three switches are turned on. 4. Exected Values of Random Variables Because a robabilit can be thought of as the long-run relative frequenc of occurrence for an event, a robabilit distribution can be interreted as showing the long-run relative frequenc of occurrences for numerical outcomes associated with an exeriment. Suose, for examle, that ou and a friend are matching balanced coins. Each of ou flis a coin. If the uer faces match, ou win $.00; if the do not match, ou lose $.00 (our friend wins $.00). The robabilit of a match is 0.5 and, in the long run, ou should win about half of the time. Thus, a relative frequenc distribution of our winnings should look like the one shown in Figure 4.4. The - under the left most bar indicates a loss of $.00 b ou. Figure 4.4. Relative frequenc of winnings

12 On average, how much will ou win er game over the long run? If Figure 4.4 resents a correct disla of our winnings, ou win - (lose a dollar) half of the time and half of the time, for an average of ( ) () 0 This average is sometimes called our exected winnings er game, or the exected value of our winnings. (A game that has an exected value of winnings of 0 is called a fair game.) The general definition of exected value is given in Definition 4.4. Definition 4.4. The exected value of a discrete random variable with robabilit distribution (x) is given b E ( ) x( x) x (The sum is over all values of x for which (x) > 0.) We sometimes use the notation E() μ for this equivalence. Note: We assume absolute convergence when the range of is countable; we talk about an exectation onl when it is assumed to exist... Now ada has arrived, and ou and our friend u the stakes to $0 er game of matching coins. You now win -0 or 0 with equal robabilit. Your exected winnings er game is ( 0) (0) 0 and the game is still fair. The new stakes can be thought of as a function of the old in the sense that, if reresents our winnings er game when ou were laing for $.00, then 0 reresents our winnings er game when ou la for $0.00. Such functions of random variables arise often. The extension of the definition of exected value to cover these cases is given in Theorem 4.. Theorem 4.. If is a discrete random variable with robabilit distribution (x) and if g(x) is an real-valued function of, then E ( g( )) g( x) ( x) x (The roof of this theorem will not be given.) You and our friend decide to comlicate the aoff rules to the coin-matching game b agreeing to let ou win $ if the match is tails and $ if the match is heads. You

13 3 still lose $ if the coins do not match. Quickl ou see that this is not a fair game, because our exected winnings are ( ) () () You comensate for this b agreeing to a our friend $.50 if the coins do not match. Then, our exected winnings er game are (.5) () () and the game is again fair. What is the difference between this game and the original one, in which all aoffs were $? The difference certainl cannot be exlained b the exected value, since both games are fair. You can win more, but also lose more, with the new aoffs, and the difference between the two games can be exlained to some extent b the increased variabilit of our winnings across man games. This increased variabilit can be seen in Figure 4.5, which dislas the relative frequenc for our winnings in the new game; the winnings are more sread out than the were in Figure 4.4. Formall, variation is often measured b the variance and b a related quantit called the standard deviation. Figure 4.5. Relative frequenc of winnings Definition 4.5. The variance of a random variable with exected value μ is given b V ( ) E[ ( μ) ]. We sometimes use the notation for this equivalence. E [( μ ) ] σ Notice that the variance can be thought of as the average squared distance between values of and the exected value μ. Thus, the units associated with σ are the square of the units of measurement for. The smallest value that σ can assume is zero. The variance is zero when all the robabilit is concentrated at a single oint, that is, when takes on a constant value with robabilit. The variance becomes larger as the oints with ositive robabilit sread out more.

14 4 The standard deviation is a measure of variation that maintains the original units of measure, as oosed to the squared units associated with the variance. Definition 4.6. The standard deviation of a random variable is the square root of the variance and is given b σ σ E [( μ) ] is For the game reresented in Figure 4.4, the variance of ou winnings (with μ 0) σ [( ) ] E μ ( ) It follows that σ, as well. For the game reresented in Figure 4.5, the variance of our winnings is σ (.5) and the standard deviation is σ σ Which game would ou rather la? The standard deviation can be thought of as the size of a tical deviation between an observed outcome and the exected value. For the situation dislaed in Figure 4.4, each outcome (- or ) deviates b recisel one standard deviation from the exected value. For the situation described in Figure 4.5, the ositive values average.5 units from the exected value of 0 (as do the negative values), and so.5 units is aroximatel one standard deviation here. The mean and the standard deviation often ield a useful summar of the robabilit distribution for a random variable that can assume man values. An illustration is rovided b the age distribution of the U.S. oulation for 000 and 00 (rojected, as shown in Table 4.). Age is actuall a continuous measurement, but since it is reorted in categories, we can treat it as a discrete random variable for uroses of aroximating its ke function. To move from continuous age intervals to discrete age classes, we assign each interval the value of its midoint (rounded). Thus, the data in Table 4. are interreted as showing that 6.9% of the 000 oulation were around 3 ears of age and that.6% of the 00 oulation is anticiated to be around 45 ears of age. (The oen intervals at the uer end were stoed at 00 for convenience.)

15 5 Table 4.. Age Distribution in 000 and 00 (Projected) Age Interval Age Midoint Under and over *Source: U.S. Census Bureau Interreting the ercentages as robabilities, we see that the mean age for 000 is aroximated b μ x( x) x 3(0.069) 8(0.073) 5(0.44)... 90(0.033) 36.6 (How does this comare with the median age for 000, as aroximated from Table 4..) For 00, the mean age is aroximated b μ x( x) x 3(0.06) 8(0.06) 5(0.8)... 90(0.099) 4.5 Over the rojected eriod, the mean age increases rather markedl (as does the median age). The variations in the two age distributions can be aroximated b the standard deviations. For 000, this is σ ( x μ) ( x) x (3 36.6) (0.069) (8 36.6).6 (0.073) (5 36.6) (0.44)... ( ) (0.033) A similar calculation for the 00 data ields σ 6.3. These results are summarized in Table 4..

16 6 Table 4.. Age Distribution of U.S. Poulation Summar Mean Standard Deviation Not onl is the oulation getting older, on average, but its variabilit is increasing. What are some of the imlications of these trends? We now rovide other examles and extensions of these basic results. Examle 4.3: A deartment suervisor is considering urchasing a hotoco machine. One consideration is how often the machine will need reairs. Let denote the number of reairs during a ear. Based on ast erformance, the distribution of is shown below: Number of reairs, x 0 3 (x) What is the exected number of reairs during a ear?. What is the variance of the number of reairs during a ear? Solution:. From Definition 4.4, we see that E( ) x( x) x 0(0.) (0.3) (0.4) 3(0.).4 The hotocoier will need to be reaired an average of.4 times er ear.. From Definition 4.5, we see that V ( ) ( x μ) ( x) x (0.4) (0.) (.4) (0.3) (.4) (0.4) (3.4) (0.) 0.84 Our work in maniulating exected values can be greatl facilitated b making use of the two results of Theorem 4.. Often, g() is a linear function. When that is the case, the calculations of exected value and variance are eseciall simle.

17 7 Theorem 4.. For an random variable and constants a and b,. E ( a b) ae( ) b. F ( a b) ae( ) b Proof: We sketch a roof of this theorem for a discrete random variable having a robabilit distribution given b (x). B Theorem 4., Notice that x E( a b) x x x a ( ax b) ( x) [( ax) ( x) b( x)] ax( x) x x( x) b ae( ) b x b( x) x ( x) ( x) must equal unit. Also, b Definition 4.5, V ( a b) E[( a b) E( a b)] E[ a b ( ae( ) b)] E[ a ae( )] E a a [ ( x E( )) ] E[ ( E( )) ] a V ( ) An imortant secial case of Theorem 4. involves establishing a standardized variable. If has mean μ and standard deviation σ, then the standardized form of is given b μ Y σ Emloing Theorem 4., one can show that E(Y) 0 and V(Y ). This idea will be used often in later chaters. We illustrate the use of these results in the following examle. Examle 4.4: The deartment suervisor in Examle 4.3 wants to consider the cost of maintenance before urchasing the hotoco machine. The cost of maintenance consists of the

18 8 exense of a service agreement and the cost of reairs. The service agreement can be urchased for $00. With the agreement, the cost of each reair is $50. Find the mean and variance of the annual costs of reair for the hotocoier Solution: Recall that the of Examle 4.3 is the annual number of reairs. The annual cost of the maintenance contract is B Theorem 4., we have E(50 00) 50E( ) 00 50(.4) Thus, the manager could anticiate the average annual cost of maintenance of the hotocoier to be $70. Also, b Theorem 4., V (50 00) 50 V ( ) We will make use of this value in a later examle. (0.84) Determining the variance b Definition 4.5 is not comutationall efficient. Theorem 4. leads us to a more efficient formula for comuting the variance as given in Theorem 4.3. Theorem 4.3. If is a random variable with mean μ, then ( ) V ( ) E μ Proof: Starting with the definition of variance, we have V ( ) E E E E E [( μ) ] ( μ μ ) ( ) E( μ) E( μ ) ( ) μ μ ( ) μ

19 9 Examle 4.5: Use the result of Theorem 4.3 to comute the variance of as given in Examle 4.3. Solution: In Examle 4.3, had a robabilit distribution given b and we found that E().4. Now, E( x 0 3 (x) ) x ( x) x 0 (0.) (0.3) (0.4) 3 (0.).8 B Theorem 4.3, V ( ) E( ) μ.8 (.4) 0.84 We have comuted means and variances for a number of robabilit distributions and noted that these two quantities give us some useful information on the center and sread of the robabilit mass. Now suose that we know onl the mean and the variance for a robabilit distribution. Can we sa anthing secific about robabilities for certain intervals about the mean? The answer is es, and a useful result of the relationshi among mean, standard deviation, and relative frequenc will now be discussed. The inequalit in the statement of the theorem is equivalent to P( μ kσ < < μ kσ ) k To interret this result, let k, for examle. Then the interval from μ - σ to μ σ must contain at least /k ¼ 3/4 of the robabilit mass for the random variable. We consider more secific illustrations in the following two examles.

20 0 Examle 4.6: The dail roduction of electric motors at a certain factor averaged 0, with a standard deviation of 0.. What can be said about the fraction of das on which the roduction level falls between 00 and 40?. Find the shortest interval certain to contain at least 90% of the dail roduction levels. Theorem 4.4: Tchebsheff s Theorem. Let be a random variable with mean μ and variance σ. Then for an ositive k, ) ( k k P < σ μ Proof: We begin with the definition of V() and then make substitutions in the sum defining this quantit. Now, σ μ σ μ σ μ σ μ μ μ μ μ σ k k k k x x x x x x x x V ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( (The first sum stos at the largest value of x smaller than μ kσ, and the third sum begins at the smallest value of x larger than μ kσ; the middle sum collects the remaining terms.) Observe that the middle sum is never negative; and for both of the outside sums, ) ( σ μ k x Eliminating the middle sum and substituting for ( μ) x in the other two, we get σ μ σ μ σ σ σ k k x k x k ) ( ) ( or σ μ σ μ σ σ k k x x k ) ( ) ( or ( ) σ μ σ σ k P k. It follows that ( ) k k P σ μ or ( ) k k P < σ μ

21 Solution:. The interval from 00 to 40 reresents μ - σ to μ σ, with μ 0 and σ 0. Thus, k and 3 k 4 4 At least 75% of all das, therefore, will have a total roduction value that falls in this interval. (This ercentage could be closer to 95% if the dail roduction figures show a mound-shaed, smmetric relative frequenc distribution.). To find k, we must set ( k ) equal to 0.9 and solve for k; that is, 0.9 k 0. k k 0 k 0 The interval or or 3.6 μ 3.6σ to μ3.6σ 0 3.6(0) to 0 3.6(0) 88.4 to 5.6 will then contain at least 90% of the dail roduction levels. Examle 4.7: The annual cost of maintenance for a certain hotoco machine has a mean of $70 and a variance of $00 (see Examle 4.5). The manager wants to budget enough for maintenance that he is unlikel to go over the budgeted amount. He is considering budgeting $400 for maintenance. How often will the maintenance cost exceed this amount? Solution: First, we must find the distance between the mean and 400, in terms of the standard deviation of the distribution of costs. We have

22 400 μ σ Thus, 400 is.84 standard deviations aboe the mean. Letting k.84 in Theorem 4.4, we can find the interval μ.84σ to μ.84σ or 70.84(45.8) to 70.84(45.8) or 40 to 400 must contain at least k (.84) of the robabilit. Because the annual cost is $00 lus $50 for each reair, the annual cost cannot be less than $00. Thus, at most 0. of the robabilit mass can exceed $400; that is, the cost cannot exceed $400 more than % of the time. Examle 4.8: Suose the random variable has the robabilit mass function given in the table below. Evaluate Tchebsheff s inequalit for k. Solution: First, we find the mean of Then and x - 0 (x) /8 3/4 /8 μ x ( x ) ( )( / 8) 0(3 / 4) ( / 8) 0 x E( ) x ( x) ( ) (/ 8) 0 (3/ 4) (/ 8) / 4 Thus, the standard deviation of is x σ E ( ) μ σ σ 4 Now, for, the robabilit is within σ of μ is 4 0 4

23 3 P( μ < σ ) P( μ < ( )) P( μ < ) P( 3 4 0) B Tchebsheff s theorem, the robabilit an random variable is within σ of μ is P( μ < σ ) k Therefore, for this articular random variable, equalit holds in Tchebsheff s theorem. Thus, one cannot imrove on the bounds of the theorem. 3 4 Exercises 4.3. You are to a $.99 to la a game that consists of drawing one ticket at random from a box of unnumbered tickets. You win the amount (in dollars) of the number on the ticket ou draw. The following two boxes of numbered tickets are available. 0,, 0, 0, 0,, 4 I. II. a. Find the exected value and variance of our net gain er la with box I. b. Reeat art (a) for box II. c. Given that ou have decided to la, which box would ou choose, and wh? 4.4. The size distribution of U.S. families is shown in the table below. Number of Persons Percentage 5.7% or more. a. Calculate the mean and the standard deviation of famil size. Are these exact values or aroximations? b. How does the mean famil size comare to the median famil size? 4.5. The table below shows the estimated number of AIDS cases in the United States b age grou.

24 4 Numbers of AIDS Cases in the U.S. during 004 Age Number of Cases Under to to 4,788 5 to 9 3, to 34 4, to 39 8,03 40 to 44 8, to 49 6,45 50 to 54 3,93 55 to 59, to or older 90 Total 4,55 Source: U.S. Centers for Disease Control Let denote the age of a erson with AIDS. a. Using the mid-oint of the interval to reresent the age of all individuals in that age categor, find the aroximate robabilit distribution for b. Aroximate the mean and the standard deviation of this age distribution. c. How does the mean age comare to the aroximate median age? 4.6. How old are our drivers? The accomaning table gives the age distribution of licensed drivers in the United States. Describe this age distribution in terms of median, mean, and standard deviation. Licensed U.S. Drivers in 004 Age Number (in millions) 9 and under and over.5 Total 98.9 Source: U.S. Deartment of Transortation

25 Who commits the crimes in the United States? Although this is a ver comlex question, one wa to address it is to look at the age distribution of those who commit violent crimes. This is resented in the table below. Describe the distribution in terms of median, mean, and standard deviation. Age Percent of Violent Crimes 4 and Under and Older A fisherman is restricted to catching at most red grouer er da when fishing in the Gulf of Mexico. A field agent for the wildlife commission often insects the da s catch for boats as the come to shore near his base. He has found the number of grouer caught has the following distribution. Number of Grouer 0 Probabilit Assuming that these records are reresentative of red grouer dail catches in the Gulf, find the exected value, the variance, and the standard deviation for the individual dail catch of red grouer Aroximatel 0% of the glass bottles coming off a roduction line have serious defects in the glass. Two bottles are randoml selected for insection. Find the exected value and the variance of the number of insected bottles with serious defects Two construction contracts are to be randoml assigned to one or more of three firms I, II, and III. A firm ma receive more than one contract. Each contract has a otential rofit of $90,000. a. Find the exected otential rofit for firm I. b. Find the exected otential rofit for firms I and II together. 4.. Two balanced coins are tossed. What are the exected value and the variance of the number of heads observed?

26 6 4.. In a romotional effort, new customers are encouraged to enter an on-line sweestakes. To la, the new customer icks 9 numbers between and 50, inclusive. At the end of the romotional eriod, 9 numbers from to 50, inclusive, are drawn without relacement from a hoer. If the customer s 9 numbers match all of those drawn (without concern for order), the customer wins $5,000,000. a. What is the robabilit that the new customer wins the $5,000,000? b. What is the exected value and variance of the winnings? c. If the new customer had to mail in the icked numbers, assuming that the cost of ostage and handling is $0.50, what is the exected value and variance of the winnings? 4.3. The number of equiment breakdowns in a manufacturing lant is closel monitored b the suervisor of oerations, since it is critical to the roduction rocess. The number averages 5 er week, with a standard deviation of 0.8 er week. a. Find an interval that includes at least 90% of the weekl figures for number of breakdowns. b. The suervisor romises that the number of breakdowns will rarel exceed 8 in a oneweek eriod. Is the director safe in making this claim? Wh? 4.4. Keeing an adequate sul of sare arts on hand is an imortant function of the arts deartment of a large electronics firm. The monthl demand for 00-gigabte hard drives for ersonal comuters was studied for some months and found to average 8 with a standard deviation of 4. How man hard drives should be stocked at the beginning of each month to ensure that the demand will exceed the sul with a robabilit of less than 0.0? 4.5. An imortant feature of golf cart batteries is the number of minutes the will erform before needing to be recharged. A certain manufacturer advertises batteries that will run, under a 75-am discharge test, for an average of 5 minutes, with a standard deviation of 5 minutes. a. Find an interval that contains at least 90% of the erformance eriods for batteries of this te. b. Would ou exect man batteries to die out in less than 00 minutes? Wh? 4.6. Costs of equiment maintenance are an imortant art of a firm s budget. Each visit b a field reresentative to check out a malfunction in a certain machine used in a manufacturing rocess is $65, and the arts cost, on average, about $5 to correct each malfunction. In this large lant, the exected number of these machine malfunctions is aroximatel 5 er month, and the standard deviation of the number of malfunctions is. a. Find the exected value and standard deviation of the monthl cost of visits b the field reresentative. b. How much should the firm budget er month to ensure that the costs of these visits are covered at least 75% of the time? At this oint, it ma seem that ever roblem has its own unique robabilit distribution, and that we must start from basics to construct such a distribution each time

27 7 a new roblem comes u. Fortunatel, this is not the case. Certain basic robabilit distributions can be develoed as models for a large number of ractical roblems. In the remainder of this chater, we shall consider some fundamental discrete distributions, looking at the theoretical assumtions that underlie these distributions as well as at the means, variances, and alications of the distributions. 4.3 The Bernoulli Distribution Numerous exeriments have two ossible outcomes. If an item is selected from the assembl line and insected, it will be found to be either defective or not defective. A iece of fruit is either damaged or not damaged. A cow is either regnant or not regnant. A child will be either female or male. Such exeriments are called Bernoulli trials after the Swiss mathematician Jacob Bernoulli. For simlicit, suose one outcome of a Bernoulli trial is identified to be a success and the other a failure. Define the random variable as follows:, if the outcome of the trial is a success 0, if the outcome of the trial is a failure If the robabilit of observing a success is, the robabilit of observing failure is. The robabilit distribution of, then, is given b ( x) x ( ) x, x 0, where (x) denotes the robabilit that x. Such a random variable is said to have a Bernoulli distribution or to reresent the outcome of a single Bernoulli trial. A general formula for (x) identifies a famil of distributions indexed b certain constants called arameters. For the Bernoulli distribution, the robabilit of success,, is the onl arameter. Suose that we reeatedl observe the outcomes of random exeriments of this te, recording a value of for each outcome. What average of should we exect to see? B Definition 4.4, the exected value of is given b E( ) x( x) x 0 (0) ( () 0( ) ( ) Thus, if we insect a single item from an assembl line and 0% of the items are defective, we should exect to observe an average of 0. defective items er item insected. (In other words, we should exect to see one defective item for ever ten items insected.) For the Bernoulli random variable, the variance (see Theorem 4.3) is

28 8 V ( ) E( x x ) [ E( )] ( x) 0 ( ) ( ) ( ) Seldom is one interested in observing onl one outcome of a Bernoulli trial. However, the Bernoulli random variable will be used as a building block to form other robabilit distributions, such as the binomial distribution of Section 4.4. The roerties of the Bernoulli distribution are summarized below. The Bernoulli Distribution ( x) x ( ) x, x 0, for 0 E ( ) and V ( ) ( ) 4.4 The Binomial Distribution 4.4. Probabilit Function Suose we conduct n indeendent Bernoulli trials, each with a robabilit of success. Let the random variable Y be the number of successes in the n trials. The distribution of Y is called the binomial distribution. As an illustration, instead of insecting a single item, as we do with a Bernoulli random variable, suose that we now indeendentl insect n items and record values for,,, n, where i if the ith insected item is defective and i 0, otherwise. The sum of the i s, Y n i i denotes the number of defectives among the n samled items. We can easil find the robabilit distribution for Y under the assumtion that P( i ), where remains constant over all trials. For the sake of simlicit, let us look at the secific case of n 3. The random variable Y can then take on four ossible values: 0,,, and 3. For Y to be 0, all three i values must be 0. Thus, P( Y 0) P( P( ( ) 0, 0) P( 0, 3 0) P( 0) 3 0) Now if Y, then exactl one value of i is and the other two are 0. The one defective could occur on an of the three trials; thus,

29 9 P( Y ) P[( P( P(,, P( 3( ) ( ) P( 0, 0, 0, ( ) 3 3 P( 0, 0, 3 0) ( because the three ossibilities are mutuall exclusive) ( ) 0) P( 0) P( 0) ( 0, 3 ( ) 3 0) P( 0) P( )] 3 0, 0) P( 0, ),, 0) P( 3 0) 3 0) ) P( 3 0) Notice that the robabilit of each secific outcome is the same, ( - ). For Y, two values of i must be and one must be 0, which can occur in three mutuall exclusive was. Hence, P( Y ) P[( P( 3, ) P( P( ( ) (, 0, 3 0) P( 0) P( ( ) ( ) 0) (, 3 0) P( 0) P( 3 ( ), )] 3 ) 0 0) P( 3 ) ) P( 3 0) The event Y 3 can occur onl if all values of i are, so P( Y 3) P( P( 3, ) P(, 3 ) P( ) 3 ) Notice that the coefficient in each of the exressions for P(Y ) is the number of was of selecting ositions, in sequence, in which to lace s. Because there are three 3 ositions in the sequence, this number amounts to. Thus we can write 3 3 P( Y ) ( ), 0,,, 3, when n 3 For general values of n, the robabilit that Y will take on a secific value sa, n is given b the term ( ) multilied b the number of ossible outcomes that result in exactl defectives being observed. This number, which reresents the number

30 30 of ossible was of selecting ositions for defectives in the n ossible ositions of the sequence, is given b n n!!( n )! where n! n( n )... and 0!. Thus, in general, the robabilit mass function for the binomial distribution is n n P( Y ) ( ) ( ), 0,,,.., n Once n and are secified we can comletel determine the robabilit function for the binomial distribution; hence, the arameters of the binomial distribution are n and. The shae of the binomial distribution is affected b both arameters n and. If 0.5, the distribution is smmetric. If < 0.5, the distribution is skewed right, becoming less skewed as n increases. Similarl, if > 0.5, the distribution is skewed left and becomes less skewed as n increases (see Figure 4.6). You can exlore the shae of the binomial distribution using the grahing binomial alet. When n, ( ) ( ) ( ), 0, the robabilit function of the Bernoulli distribution. Thus, the Bernoulli distribution is a secial case of the binomial distribution with n. Figure 4.6. Binomial robabilities for < 0.5, 0.5, > 0.5 Notice that the binomial robabilit function satisfies the two conditions of a robabilit function. First, robabilities are nonnegative. Second, the sum of the robabilities is one, which can be verified using the binomial theorem: n n n ( x) ( ) x x 0 n ( ( ))

31 3 Although we have used to denote the robabilit of success, q is a common notation that we will use here and in later sections. To summarize, a random variable Y ossesses a binomial distribution if the following conditions are satisfied:. The exeriment consists of a fixed number n of identical trials.. Each trial can result in one of onl two ossible outcomes, called success or failure; that is, each trial is a Bernoulli trial. 3. The robabilit of success is constant from trial to trial. 4. The trials are indeendent. 5. Y is defined to be the number of successes among the n trials. Man exerimental situations involve random variables that can be adequatel modeled b the binomial distribution. In addition to the number of defectives in a samle of n items, examles include the number of emloees who favor a certain retirement olic out of n emloees interviewed, the number of istons in an eight-clinder engine that are misfiring, and the number of electronic sstems sold this week out of the n that were manufactured. Examle 4.9: Suose that 0% of a large lot of ales are damaged. If four ales are randoml samled from the lot, find the robabilit that exactl one ale is damaged. Find the robabilit that at least one ale in the samle of four is defective. Solution: We assume that the four trials are indeendent and that the robabilit of observing a damaged ale is the same (0.) for each trial. This would be aroximatel true if the lot indeed were large. (If the lot contained onl a few ales, removing one ale would substantiall change the robabilit of observing a damaged ale on the second draw.) Thus, the binomial distribution rovides a reasonable model for this exeriment, and we have (with Y denoting the number of defectives) 4 3 () (0.) (0.9) 0.96 To find P(Y ), we observe that P( Y ) P( Y 4 (0.) 0 (0.9) ) (0) 4 0 (0.9) 4

32 3 Discrete distributions, like the binomial, can arise in situations where the underling roblem involves a continuous (that is, nondiscrete) random variable. The following examle rovides an illustration. Examle 4.0: In a stud of life lengths for a certain batter for lato comuters, researchers found that the robabilit that a batter life will exceed 5 hours is 0.. If three such batteries are in use in indeendent latos, find the robabilit that onl one of the batteries will last 5 hours or more. Solution: Letting Y denote the number of batteries lasting 5 hours or more, we can reasonabl assume Y to have a binomial distribution, with 0.. Hence, 3 P ( Y ) () (0.) (0.88) Mean and Variance There are numerous was of find E(Y) and V(Y) for a binomiall distributed random variable Y. We might use the basic definition and comute E( Y ) n 0 ( ) n ( ) but direct evaluation of this exression is a bit trick. Another aroach is to make use of the results on linear functions of random variables, which will be resented in Chater 6. We shall see in Chater 6 that, because the binomial Y arose as a sum of indeendent Bernoulli random variables,,, n, n and E( Y ) E n i 0 n i n n i i E( ) i

33 33 V ( Y ) n V ( ) i i i n ( ) n( ) Examle 4.: Referring to examle 4.9, suose that a customer is the one who randoml selected and then urchased the four ales. If an ale is damaged, the customer comlains. To kee the customers satisfied, the store has a olic of relacing an damaged item (here the ale) and giving the customer a couon for future urchases. The cost of this rogram has, through time, been found to be C 3Y, where Y denotes the number of defective ales in the urchase of four. Find the exected cost of the rogram when a customer randoml selects 4 ales from the lot. Solution: We know that E ( C) E ( 3Y ) 3E( Y ) and it now remains for us to find E(Y ). From Theorem 4.3, V ( ) ( Y ) E( Y μ) E Y μ Since V ( Y ) n( ) and μ E ( Y ) n, we see that ( ) E Y For examle 4.9, 0. and n 4; hence, V ( Y ) μ n( ) ( n) E( Y ) 3[ n( q ) ( ) ] [ (4) (0.) ] E( C) 3 n 3 4(0.)(0.9).56 If the costs were originall exressed in dollars, we could exect to a an average of $.56 when a customer urchases 4 ales Histor and Alications The binomial exansion can be written as

34 34 n n n x n x ( a b) a b x 0 x If a, where 0 < <, and b, we see that the terms on the left are the robabilities of the binomial distribution. Long ago it was found that the binomial n coefficients,, could be generated from Pascal s triangle (see Figures 4.7 and 4.8). x Figure 4.7. Blaise Pascal (63 66) Source: htt://oregonstate.edu/instruct/hl30/hilosohers/ascal.html Figure 4.8. Pascal s triangle To construct the triangle, the first two rows are created, consisting of s. Subsequent rows have the outside entries as ones; each of the interior numbers is the sum of the numbers immediatel to the left and to the right on the row above. According to David (955), the Chinese writer Chu Shih-chieh ublished the arithmetical triangle of binomial coefficients in 303, referring to it as an ancient method.

35 35 The triangle seems to have been discovered and rediscovered several times. Michael Stifel ublished the binomial coefficients in 544 (Eves 969). Pascal s name seems to have become firml attached to the arithmetical triangle, becoming known as Pascal s triangle, about 665, although a triangle-te arra was also given b Bernoulli in the 73 Ars Conjectandi. Jacob Bernoulli ( ) is generall credited for establishing the binomial distribution for use in robabilit (see Figure 4.9) (Folks 98, Stigler 986). Although his father lanned for him to become a minister, Bernoulli became interested in and began to ursue mathematics. B 684, Bernoulli and his ounger brother John had develoed differential calculus from hints and solutions rovided b Leibniz. However, the brothers became rivals, corresonding in later ears onl be rint. Jacob Bernoulli would ose a roblem in a journal. His brother John would rovide an answer in the same issue, and Jacob would resond, again in rint, that John had made an error. Figure 4.9. Jacob Bernoulli ( ) Source: htt:// When Bernoulli died of a slow fever on August 6, 795, he left behind numerous unublished, and some uncomleted, works. The most imortant of these was on robabilit. He had worked over a eriod of about twent ears rior to his death on the determination of chance, and it was this work that his nehew ublished in 73, Ars Conjectandi (The Art of Conjecturing). In this work, he used the binomial exansion to address robabilit roblems, resented his theor of ermutations and combinations, develoed the Bernoulli numbers, and rovided the weak law of large numbers for Bernoulli trials. As was the case with man of the earl works in robabilit, the earl develoments of the binomial distribution resulted from efforts to address questions relating to games of chance. Subsequentl, roblems in astronom, the social sciences, insurance, meteorolog, and medicine are but a few of those that have been addressed using this distribution. Polls are frequentl reorted in the newsaer and on radio and television. The binomial distribution is used to determine how man eole to surve and how to resent the results. Whenever an event has two ossible outcomes and n such events are to be observed, the binomial distribution is generall the first model considered. This has led it to be widel used in qualit control in manufacturing rocesses.

36 36 Determining the robabilities of the binomial distribution quickl becomes too comlex to be done quickl b hand. Man calculators and software rograms have built-in functions for this urose. Table in the Aendix gives cumulative binomial robabilities for selected values of n and. The entries in the table are values of a n n n ( ) ( ), a 0,,, n 0 0 The following examle illustrates the use of the table. Examle 4.: An industrial firm sulies ten manufacturing lants with a certain chemical. The robabilit that an one firm will call in an order on a given da is 0., and this robabilit is the same for all ten lants. Find the robabilit that, on the given da, the number of lants calling in orders is as follows.. at most 3. at least 3 3. exactl 3 Solution: Let Y denote the number of lants that call in orders on the da in question. If the lants order indeendentl, then Y can be modeled to have a binomial distribution with 0... We then have P( Y 3) 3 0 ( ) (0.) (0.8) This ma be determined using Table (b) in the Aendix. Note we use (b) because it corresonds to n 0. Then the robabilit corresonds to the entr in column 0. and row k 3. Notice the binomial calculator alet rovides the same result.. Notice that P( Y 3) P( Y ) 0 0 (0.) (0.8) Here we took advantage of the fact that ositive robabilit onl occurs at integer values so that, for examle, P ( Y.5) Observe that 0

37 37 P( Y from the results just established. 3) P( Y 3) P( Y ) The examles used to this oint have secified n and in order to calculate robabilities or exected values. Sometimes, however, it is necessar to choose n so as to achieve a secified robabilit. Examle 4.3 illustrates the oint. Examle 4.3: Ever hosital has backu generators for critical sstems should the electricit go out. Indeendent but identical backu generators are installed so that the robabilit that at least one sstem will oerate correctl when called uon is no less than Let n denote the number of backu generators in a hosital. How large must n be to achieve the secified robabilit of at least one generator oerating, if. 0.95?. 0.8? Solution: Let Y denote the number of correctl oerating generators. If the generators are identical and indeendent, Y has a binomial distribution. Thus, P( Y ) P( Y 0) n 0 ( ) 0 ( ) n n The conditions secif that n must be such that P ( Y ) or more.. When 0.95, n P ( Y ) ( 0.95) 0.99 results in n (0.05) 0.99 or n ( 0.05) so n ; that is, installing two backu generators will satisf the secifications.. When 0.80, n P ( Y ) ( 0.8) 0.99 results in

38 38 n ( 0.) 0.0 Now (0.) 0.04, and (0.) , so we must go to n 3 sstems to ensure that 3 P ( Y ) (0.) 0.99 > 0.99 Note: We cannot achieve the 0.99 robabilit exactl, because Y can assume onl integer values. Examle 4.4: Virtuall an rocess can be imroved b the use of statistics, including the law. A much-ublicized case that involved a debate about robabilit was the Collins case, which began in 964. An incident of urse snatching in the Los Angeles area led to the arrest of Michael and Janet Collins. At their trial, an exert resented the following robabilities on characteristics ossessed b the coule seen running from the crime. The chance that a coule had all of these characteristics together is in million. Since the Collinses had all of the secified characteristics, the must be guilt. What, if anthing, is wrong with this line of reasoning? Man with beard 0 Blond woman 4 Yellow car 0 Woman with ontail 0 Man with mustache 3 Interracial coule 000 Solution: First, no background data are offered to suort the robabilities used. Second, the six events are not indeendent of one another and, therefore, the robabilities cannot be multilied. Third, and most interesting, the wrong question is being addressed. The question of interest is not What is the robabilit of finding a coule with these characteristics? Since one such coule has been found (the Collinses), the roer question is: What is the robabilit that another such coule exists, given that we found one? Here is where the binomial distribution comes into la. In the binomial model, let n Number of coules who could have committed the crime Probabilit that an one coule ossesses the six listed characteristics x Number of coules who ossess the six characteristics From the binomial distribution, we know that

39 39 P( P( P( 0) ( ) n ) n( ) n ) ( ) n Then, the answer to the conditional question osed above is P( > P[( > ) ( ) P( ) P( P( > ) ) )] n ( ) n( ) n ( ) n Substituting / million and n million, which are lausible but not welljustified guesses, we get P ( > ) 0.4 so the robabilit of seeing another such coule, given that we have alread seen one, is much larger than the robabilit of seeing such a coule in the first lace. This holds true even if the numbers are dramaticall changed. For instance, if n is reduced to million, the conditional robabilit becomes 0.05, which is still much larger than / million. The imortant lessons illustrated here are that the correct robabilit question is sometimes difficult to determine and that conditional robabilities are ver sensitive to conditions. We shall soon move on to a discussion of other discrete random variables; but the binomial distribution, summarized below, will be used frequentl throughout the remainder of the text. The Binomial Distribution n n ( ) ( ), 0,,,.., n for 0 E ( Y ) n V ( Y ) n( ) Exercises 4.7. Let denote a random variable that has a binomial distribution with 0.3 and n 5. Find the following values. a. P( 3) b. P( 3)

40 40 c. P( 3) d. E() e. V() 4.8. Let denote a random variable that has a binomial distribution with 0.6 and n 5. Use our calculator, Table in the Aendix, or the binomial calculator alet to evaluate the following robabilities. a. P( 0) b. P( 5) c. P( 0) 4.9. A machine that fills milk cartons underfills a certain roortion. If 50 boxes are randoml selected from the outut of this machine, find the robabilit that no more than cartons are underfilled when a b When testing insecticides, the amount of the chemical when, given all at once, will result in the death of 50% of the oulation is called the LD50, where LD stands for lethal dose. If 40 insects are laced in searate Petri dishes and treated with an insecticide dosage of LD50, find the robabilities of the following events. a. Exactl 0 survive b. At most 5 survive c. At least 0 survive 4.3. Refer to Exercise a. Find the number exected to survive, out of 40. b. Find the variance of the number of survivors, out of Among ersons donating blood to a clinic, 85% have Rh blood (that is, the Rhesus factor is resent in their blood.) Six eole donate blood at the clinic on a articular da. a. Find the robabilit that at least one of the five does not have the Rh factor. b. Find the robabilit that at most four of the six have Rh blood The clinic in Exercise 4.3 needs six Rh donors on a certain da. How man eole must donate blood to have the robabilit of obtaining blood from at least six Rh donors over 0.95? During the 00 Surve of Business Owners (SBO), it was found that the numbers of female-owned, male-owned, and jointl male- and female-owned business were 6.5, 3., and.7 million, resectivel. Among four randoml selected businesses, find the robabilities of the following events. a. All four had a female, but no male, owner. b. One of the four was either owned or co-owned b a male. c. None of the four were jointl owned b female and a male.

41 Goranson and Hall (980) exlain that the robabilit of detecting a crack in an airlane wing is the roduct of, the robabilit of insecting a lane with a wing crack;, the robabilit of insecting the detail in which the crack is located; and 3, the robabilit of detecting the damage. a. What assumtions justif the multilication of these robabilities? b. Suose that 0.9, 0.8, and for a certain fleet of lanes. If three lanes are insected from this fleet, find the robabilit that a wing crack will be detected in at least one of them Each da a large animal clinic schedules 0 horses to be tested for a common resirator disease. The cost of each test is $80. The robabilit of a horse having the disease is 0.. If the horse has the disease, treatment costs $500. a. What is the robabilit that at least one horse will be diagnosed with the disease on a randoml selected da? b. What is the exected dail revenue that the clinic earns from testing horses for the disease and treating those who are sick? The efficac of the mums vaccine is about 80%; that is, 80% of those receiving the mums vaccine will not contract the disease when exosed. Assume each erson s resonse to the mums is indeendent of another erson s resonse. Find the robabilit that at least one exosed erson will get the mums if n are exosed where a. n b. n Refer to Exercise a. How man vaccinated eole must be exosed to the mums before the robabilit that at least one erson will contract the disease is at least 0.95? b. In 006, an outbreak of mums in Iowa resulted in 605 susect, robable, and confirmed cases. Given broad exosure, do ou find this number to be excessivel large? Justif our answer A comlex electronic sstem is built with a certain number of backu comonents in its subsstems. One subsstem has four identical comonents, each with a robabilit of 0.5 of failing in less than 000 hours. The subsstem will oerate if an two or more of the four comonents are oerating. Assuming that the comonents oerate indeendentl, find the robabilities of the following events. a. Exactl two of the four comonents last longer than 000 hours. b. The subsstem oerates for longer than 000 hours In a stud, dogs were trained to detect the resence of bladder cancer b smelling urine (USA Toda, Setember 4, 004). During training, each dog was resented with urine secimens from health eole, those from eole with bladder cancer, and those from eole sick with unrelated diseases. The dog was trained to lie down b an urine secimen from a erson with bladder cancer. Once training was comleted, each dog was resented with seven urine secimens, onl one of which came from a erson with

42 4 bladder cancer. The secimen that the dog laid down beside was recorded. Each dog reeated the test nine times. Six dogs were tested. a. One dog had onl one success in 9. What is the robabilit of the dog having at least this much success if it cannot detect the resence of bladder cancer b smelling a erson s urine? b. Two dogs correctl identified the bladder cancer secimen on 5 of the 9 trials. If neither were able to detect the resence of bladder cancer b smelling a erson s urine, what is the robabilit that both dogs correctl detected the bladder secimen on at least 5 of the 9 trials? 4.4. A firm sells four items randoml selected from a large lot that is known to contain % defectives. Let Y denote the number of defectives among the four sold. The urchaser of the items will return the defectives for reair, and the reair cost is given b C Y Y 3 Find the exected reair cost From a large lot of memor chis for use in ersonal comutes, n are to be samled b a otential buer, and the number of defectives is to be observed. If at least one defective is observed in the samle of n, the entire lot is to be rejected b the otential buer. Find n so that the robabilit of detecting at least one defective is aroximatel 0.95 if the following ercentages are correct. a. 0% of the lot is defective. b. 5% of the lot is defective Fifteen free-standing ranges with smoothtos are available for sale in a wholesale aliance dealer s warehouse. The ranges sell for $550 each, but a double-our-moneback guarantee is in effect for an defective range the urchaser might urchase. Find the exected net gain for the seller if the robabilit of an one range being defective is (Assume that the qualit of an one range is indeendent of the qualit of the others.) 4.5 The Geometric Distribution 4.5. Probabilit Function Suose that a series of test firings of a rocket engine can be reresented b a sequence of indeendent Bernoulli random variables, with i if the ith trial results in a successful firing and with i 0, otherwise. Assume that the robabilit of a successful firing is constant for the trials, and let this robabilit be denoted b. For this roblem, we might be interested in the number of failures rior to the trial on which the first successful firing occurs. If Y denotes the number of failures rior to the first success, then

43 43 P( Y ) ( ) P( P( ( )( ) ( ) ( ) q 0) P( 0, 0,..., 0) P(, 0,,,... 0, 0) P( ) ) because of the indeendence of the trials. This formula is referred to as the geometric robabilit distribution. Notice that this random variable can take on a countabl infinite number of ossible values. In addition, P( Y ) q q < P( Y [ q ] qp( Y ) ),,,... That is, each succeeding robabilit is less than the revious one (see Figure 4.0). Figure 4.0. Geometric distribution robabilit function In addition to the rocket-firing examle just given, other situations ma result in a random variable whose robabilit can be modeled b a geometric distribution: the number of customers contacted before the first sale is made; the number of ears a dam is in service before it overflows; and the number of automobiles going through a radar check before the first seeder is detected. The following examle illustrates the use of the geometric distribution. Examle 4.5: A recruiting firm finds that 0% of the alicants for a articular sales osition are fluent in both English and Sanish. Alicants are selected at random from the ool and interviewed sequentiall. Find the robabilit that five alicants are interviewed before finding the first alicant who is fluent in both English and Sanish.

44 44 Solution: Each alicant either is or is not fluent in English and Sanish, so the interview of an alicant corresonds to a Bernoulli trial. The robabilit of finding a suitable alicant will remain relativel constant from trial to trial if the ool of alicants is reasonabl large. Because alicants will be interviewed until the first one fluent in English and Sanish is found, the geometric distribution is aroriate. Let Y the number of unqualified alicants rior to the first qualified one. If five unqualified alicants are interviewed before finding the first alicant who is fluent in English and Sanish, we want to find the robabilit that Y 5. Thus, (0.) (0.8) (5) 5) ( 5 Y P The name of the geometric distribution comes from the geometric series its robabilities reresent. Proerties of the geometric series are useful when finding the robabilities of the geometric distribution. For examle, the sum of a geometric series is t t x x 0 for t <. Using this fact, we can show the geometric robabilities sum to one: ) ( ) ( ) ( ) ( 0 0 Similarl, using the artial sum of a geometric series, we can find the functional form of the geometric distribution function. For an integer 0, 0 0 ) ( ) ( t t t t q q q q q q Y P F

45 45 Using the distribution function, we have, for an integer 0, ) ( ) ( ) ( > q q F Y P Mean and Variance From the basic definition, [ ] [ ] ) ( ) ( q q q q q q q q Y E The infinite series can be slit u into a triangular arra of series as follows: ] [ ) ( q q q q q q Y E Each line on the right side is an infinite, decreasing geometric rogression with common ratio q. Recall that ) ( x a ax ax a if x <. Thus, the first line inside the bracket sums to /( q) /; the second, to q/; the third, to q /; and so on. On accumulating these totals, we then have [ ] q q q q q q q q q Y E ) ( This answer for E(Y) should seem intuitivel realistic. For examle, if 0% of a certain lot of items are defective, and if an insector looks at randoml selected items one at a time, she should exect to find nine good items before finding the first defective one. The variance of the geometric distribution will be derived in Section 4.4 and in Chater 6. The result, however, is

46 46 V ( Y ) q Examle 4.6: Referring to examle 4.5, let Y denote the number of unqualified alicants interviewed rior to the first qualified one. Suose that the first alicant fluent in both English and Sanish is offered the osition, and the alicant accets. If each interview costs $5, find the exected value and the variance of the total cost of interviewing until the job is filled. Within what interval should this cost be exected to fall? Solution: Because (Y ) is the number of the trial on which the interviewing rocess ends, the total cost of interviewing is C 5(Y ) 5Y 5. Now, and E( C) 5E( Y ) 5 q V ( C) (5) V ( Y ) q (0.) 500 The geometric is the onl discrete distribution that has the memorless roert. B this we mean that, if we know the number of failures exceed j, the robabilit that there will be more than j k failures rior to the first success is equal to the robabilit that the number of failures exceeds k; that is, for integers j and k greater than 0, P ( Y > j k Y > j) P( Y > k) To verif this, we will use the roerties of conditional robabilities.

47 47 P( Y > j k Y > P j) (( Y > j k) ( Y > j) ) P( Y P( Y > j k) P( Y > j) > j) j q q k j q k P( Y > k) Examle 4.7: Referring once again to Examle 4.5, suose that 0 alicants have been interviewed and no erson fluent in both English and Sanish have been identified. What is the robabilit that 5 unqualified alicants will be interviewed before finding the first alicant who is fluent in English and Sanish? Solution: B the memorless roert, the robabilit that fifteen unqualified alicants will be interviewed before finding an alicant who is fluent in English and Sanish, given the first ten are not qualified is equal to the robabilit of finding the first qualified candidate after interviewing 5 unqualified alicants. Again, let Y denote the number of unqualified alicants interviewed rior to the first candidate who is fluent in English and Sanish. Thus, P( ( Y 5) ( Y 0) ) P( Y 5 Y 0) P( Y 0) P( Y 5) P( Y > 9) 5 q 0 q 5 q P( Y 5)

48 48 The Geometric Distribution ( ) ( ), 0,,,.. for 0 E Y ) q q V ( Y ) ( 4.6 The Negative Binomial Distribution Probabilit Function In section 4.5, we saw that the geometric distribution models the robabilistic behavior of the number of failures rior to the first success in a sequence of indeendent Bernoulli trials. But what if we were interested in the number of failures rior to the second success, or the third success, or (in general) the rth success. The distribution governing robabilistic behavior in these cases is called the negative binomial distribution. Let Y denote the number of failures rior to the rth success in a sequence of indeendent Bernoulli trials, with denoting the common robabilit of success. We can derive the distribution of Y from known facts. Now, P(Y ) P( st ( r ) trials contain (r ) successes and the ( r)th trial is a success) P[ st trials contain (r ) successes] x P[th trial is a success] Because the trials are indeendent, the joint robabilit can be written as a roduct of robabilities. The first robabilit statement is identical to the one that results in a binomial model; and hence, P( Y ) ( ) r r ( ) r r r q, 0,,... r Notice that, if r, we have the geometric distribution. Thus, the geometric distribution is a secial case of the negative binomial distribution. The negative binomial is a quite flexible model. Its shae ranges from one being highl skewed to the right when r is small to one that is relativel smmetric as r becomes large and small. Some of these are dislaed in Figure 4..

49 49 Figure 4.. Form of the negative binomial distribution Examle 4.8: As in Examle 4.5, 0% of the alicants for a certain sales osition are fluent in English and Sanish. Suose that four jobs requiring fluenc in English and Sanish are oen. Find the robabilit that two unqualified alicants are interviewed before finding the fourth qualified alicant, if the alicants are interviewed sequentiall and at random. Solution: Again we assume indeendent trials, with 0. being the robabilit of finding a qualified candidate on an one trial. Let Y denote the number of unqualified alicants interviewed rior to interviewing the 4 th alicant who is fluent in English and Sanish. Y can reasonabl be assumed to have a negative binomial distribution, so 5 4 P( Y ) () (0.) (0.8) 3 4 0(0.) (0.8) Mean and Variance The exected value, or mean, and the variance for the negative binomiall distributed random variable Y can easil be found b analog with the geometric distribution. Recall that Y denotes the number of the failures rior to the rth success. Let W denote the number of failures rior to the first success; let W denote the number of failures between the first success and the second success; let W 3 denote the number of failures between the second success and the third success; and so forth. The results of the trials can then be reresented as follows (where F stands for failure and S reresents a success): FF F W S FF S FF F,, W W 3 S

50 50 Clearl, Y r W i i, where the W i values are indeendent and each has a geometric distribution. Thus, b results to be derived in Chater 6, and E( Y ) V ( Y ) r E( W ) i i i r V ( W ) r i i i r q q rq rq Examle 4.9: A swimming ool reairman has three check valves in stock. Ten ercent of the service calls require a check valve. What is the exected number and variance of the number of service calls she will make before using the last check valve? Solution: Let Y denote the number of service calls that do not require a check valve that the reairman will make before using the last check valve. Assuming that each service call is indeendent from the others and that the robabilit of needing a check valve is constant for each service call, the negative binomial is a reasonable model with r 3 and 0.. The total number of service calls made to use the three check valves is C Y 3. Now, E( C) E( Y ) 3 3(0.9) and V ( C) V ( Y ) 3(0.9) Histor and Alications William S. Gosset ( ) studied mathematics and chemistr at New College Oxford before joining the Arthur Guinness Son and Coman in 899 (see Figure 4.). At the brewer, he worked on a variet of mathematical and statistical roblems that arose in the brewing rocess, ublishing the results of his efforts under the en name Student. He encountered the negative binomial while working with the distributions of

51 east cells counted with a haemoctometer (907). Gosset reasoned that, if the liquid in which the cells were susended was roerl mixed, then a given article had an equal chance of falling on an unit area of the haemoctometer. Thus, he was working with the binomial distribution, and he focused on estimating the arameters n and. To his surrise, in two of his four series, the estimated variance exceeded the mean, resulting in negative estimates of n and. Nevertheless, these negative binomials fit his data well. He noted that this ma have occurred due to a tendenc of the east cells to stick together in grous which was not altogether abolished even b vigorous shaking (. 357). Several other cases aeared in the literature during the earl 900 s where estimation of the binomial arameters resulted in negative values of and n. This henomenon was exlained to some extent b arguing that for small and large n, the variabilit of the estimators would cause some negative estimates to be observed. Whitaker (95) investigated the validit of this claim. In addition to Student s work, she reviewed that of Mortara who dealt with deaths due to chronic alcoholism and that of Bortkewitsch who studied suicides of children in Prussia, suicides of women in German states, accidental deaths in trade societies, and deaths from the kick of a horse in Prussian arm cors. Whitaker found it highl unlikel that all negative estimates of and n could be exlained b variabilit. She, therefore, suggested that a new interretation was needed with the negative binomial distribution. Although we have motivated the negative binomial distribution as being the number of failures rior to the rth success in indeendent Bernoulli trials, several other models have been described that gives rise to this distribution (Boswell and Patil 970). The Póla distribution and the Pascal distribution have been other names for the negative binomial distribution. The Póla distribution was motivated from an urn model and generall refers to the secial case where r is a ositive integer, although r ma be an ositive number for the more general negative binomial distribution. 5

52 5 Figure 4.. William S. Gosset Source: htt:// Because the negative binomial distribution has been derived in so man different was, it has also been resented using different arameterizations as well. Some define the negative binomial as the number of trials required to get the first success. Because we must have at least one trial, the oints of ositive robabilit begin with, not 0, as resented earlier. The negative binomial has been used extensivel to model the number of organisms within a samling unit. For these alications, the arameters are often taken to be r and the mean μ, instead of r and, because the mean μ is of rimar interest in such studies. For these reasons, it is imortant to read carefull how the negative binomial random variable is defined when going from one source to another. The negative binomial distribution has been alied in man fields including accident statistics, oulation counts, schological data, and communications. Some of these alications will be highlighted in the exercises. The Negative Binomial Distribution r r ( ) ( ), r 0,,,.. for 0 rq rq E ( Y ) V ( Y ) Exercises Let Y denote a random variable that has a geometric distribution, with a robabilit of success on an trial denoted b. Let 0..

53 53 a. Find P(Y > ). b. Find P(Y > 4 Y > ) Let Y denote a negative binomial random variable, with 0.5. Find P(Y 4) for the following values of r. a. r b. r Suose that 0% of the engines manufactured on a certain assembl line are defective. If engines are randoml selected one at a time and tested, find the robabilit that two defective engines will be found before a good engine is found Referring to Exercise 4.46, find the robabilit that the 5 th nondefective engine will be found as follows: a. After obtaining defectives b. After obtaining 4 defectives Referring to Exercise 4.46, given that the first two engines are defective, find the robabilit that at least two more defecties are tested before the first nondefective engine is found Referring to Exercise 4.46, find the mean and the variance of the number of defectives tested before the following events occur. a. The first nondefective engine is found b. The third nondefective engine is found Greenbugs are ests in oats. If their oulations get too high, the cro will be destroed. When recording the number of greenbugs on randoml selected seedling oat lants, the counts have been found to be modeled well b the geometric distribution. Suose the average number of greenbugs on a seedling oat lant is one. Find the robabilit that a randoml selected lant has a. No greenbugs b. Two greenbugs c. At least one greenbug 4.5. The emloees of a firm that does asbestos cleanu are being tested for indications of asbestos in their lungs. The firm is asked to send four emloees who have ositive indications of asbestos on to a medical center for further testing. If 40% of the emloees have ositive indications of asbestos in their lungs, find the robabilit that six emloees who do not have asbestos in their lungs must be tested before finding the four who do have asbestos in their lungs Referring to Exercise 4.5, if each test costs $40, find the exected value and the variance of the total cost of conducting the tests to locate four ositives. Is it highl likel that the cost of comleting these tests will exceed $650?

54 Peole with O negative blood are called universal donors because the ma give blood to anone without risking incomatibilit due to the Rh factor. Nine ercent of the ersons donating blood at a clinic have O negative blood, find the robabilities of the following events. a. The first O negative donor is found after blood ting 5 eole who were not O b. The second O negative donor is the sixth donor of the da A geological stud indicates that an exlorator oil well drilled in a certain region should strike oil with robabilit 0.5. Find the robabilities of the following events. a. The first strike of oil comes after drilling three dr (non-roductive) wells b. The third strike of oil comes after three dr wells c. What assumtions must be true for our answers to be correct? In the setting of Exercies 4.5, suose that a coman wants to set u three roducing wells. Find the exected value and the variance of the number of wells that must be drilled to find three successful ones. (Hint: First find the exected value and variance of the number of dr wells that will be drilled before finding the three successful ones.) A large lot of tires contains 5% defectives. Four tires are to be chosen from the lot and laced on a car. a. Find the robabilit that two defectives are found before four good ones b. Find the exected value and the variance of the number of selections that must be made to get four good tires. (Hint: First find the exected value and variance of the number of defective tires that will be selected before finding the four good ones.) An interviewer is given a list of otential eole she can interview. Suose that the interviewer needs to interview 5 eole and that each erson indeendentl agrees to be interviewed with robabilit 0.6. Let be the number of eole she must ask to be interviewed to obtain her necessar number of interviews. a. What is the robabilit that she will be able to obtain the 5 eole b asking no more than 7 eole? b. What is the exected value and variance of the number of eole she must ask to interview 5 eole? A car salesman is told that he must make three sales each da. The salesman believes that, if he visits with a customer, the robabilit that customer will urchase a car is 0.. a. What is the robabilit that the salesman will have to visit with at least five customers to make three sales? b. What is the exected number of customers the salesman must visit with to make his dail sales goal? The number of cotton fleahoers (a est) on a cotton lant has been found to be modeled well using a negative binomial distribution with r. Suose the average

55 55 number of cotton fleahoers on lants in a cotton field is two. Find the robabilit that a randoml selected cotton lant has the following number of fleahoers. a. No cotton fleahoer b. 5 cotton fleahoers c. At least one cotton fleahoer The number of thunderstorm das in a ear has been modeled using a negative binomial model (Sakamoto 973). A thunderstorm da is defined as a da during which at least one thunderstorm cloud (cumulonimbus) occurs accomanied b lightning and thunder. It ma or ma not be accomanied b strong gusts of wind, rain, or hail. For one such site, the mean and variance of the number thunderstorm das are 5 das and 40 das, resectivel. For a randoml selected ear, find the robabilities of the following events. a. No thunderstorm das during a ear b. 0 thunderstorm das during a ear c. At least 40 thunderstorm das during a ear 4.6. Refer again to Exercise Someone considering moving to the area is concerned about the number of thunderstorm das in a ear. He wants assurances that there will be onl a 0% chance of the number of thunderstorm das exceeding a secified number of das. Find the number of das that ou ma roerl use in making this assurance This roblem is known as the Banach Match Problem. A ie-smoking mathematician alwas caries two matchboxes, one in his right-hand ocket and one in his left-hand ocket. Each time he needs a match he is equall likel to take it from either ocket. The mathematician discovers that one of his matchboxes is emt. If it is assumed that both matchboxes initiall contained N matches, what is the robabilit that there are exactl k matches in the other box, k 0,,,, N? Refer to Exercise 4.6. Suose that instead of N matches in each box, the lefthand ocket originall has N matches and the one in the right-hand ocket originall has N matches. What is the robabilit that there are exactl k matches in the other box, k 0,,,, N? 4.7. The Poisson Distribution 4.7. Probabilit Function A number of robabilit distributions come about through limiting arguments alied to other distributions. One ver useful distribution of this te is called the Poisson distribution. Consider the develoment of a robabilistic model for the number of accidents that occur at a articular highwa intersection in a eriod of one week. We can think of the time interval as being slit u into n subintervals such that

56 56 P(One accident in a subinterval) P(No accidents in a subinterval) Here we are assuming that the same value of holds for all subintervals, and that the robabilit of more than one accident occurring in an one subinterval is zero. If the occurrence of accidents can be regarded as indeendent from subinterval to subinterval, the total number of accidents in the time eriod (which equals the total number of subintervals that contain one accident) will have a binomial distribution. Although there is no unique wa to choose the subintervals and we therefore know neither n nor it seems reasonable to assume that, as n increases, should decrease. Thus, we want to look at the limit of the binomial robabilit distribution as n and 0. To get something interesting, we take the limit under the restriction that the mean (n in the binomial case) remains constant at a value we will call λ. Now with n λ or λ/n, we have Noting that n λ lim n n λ n n λ lim n! λ n λ λ lim n! n n n n n( n ) ( n ) n lim n λ n n λ λ e λ n n n n and that all other terms involving n tend toward unit, we have the limiting distribution λ λ ( ) e, 0,,,.. for λ > 0! Recall that λ denotes the mean number of occurrences in one time eriod (a week, for the examle under consideration); hence, if t non-overlaing time eriods were considered, the mean would be λt. Based on this derivation, the Poisson distribution is often referred to as the distribution of rare events. This distribution, called the Poisson distribution with arameter λ, can be used to model counts in areas of volumes, as well as in time. For examle, we ma use this distribution to model the number of flaws in a square ard of textile, the number of bacteria colonies in a cubic centimeter of water, or the number of times a machine fails in the course of a workda. We illustrate the use of the Poisson distribution in the following examle.

57 57 Examle 4.9: During business hours, the number of calls assing through a articular cellular rela sstem averages five er minute. Find the robabilit that no call wil be received during a given minute. Solution: If calls tend to occur indeendentl of one another, and if the occur at a constant rate over time, the Poisson model rovides an adequate resentation of the robabilties. Thus, 0 5 (0) e 0! 5 e The shae of the Poisson changes from highl skewed when λ is small to fairl smmetric when λ is large. (See Figure 4.3) Calculators and functions in comuter software often can be used to find the robabilities from the robabilit mass function and the cumulative distribution function. Table 3 of the Aendix gives values for cumulative Poisson robabilities of the form a 0 e λ λ,! a 0,,, for selected values of λ. Figure 4.3. Form of the Poisson distribution The following examle illustrates the use of Table 3 and the Poisson alet. Examle 4.0: Refer to Examle 4.9, and let Y denote the number of calls in the given minute. Find P(Y 4), P(Y 4), and P(Y 4).

58 58 Solution: From Table 3 or the Poisson alet, 4 (5) 5 P ( Y 4) e ! Also, P( Y 4) P( Y 3) 3 0 (5)! e and P( Y 4) P( Y 4) P( Y ) Mean and Variance We can intuitivel determine what the mean and the variance of a Poisson distribution should be b recalling the mean and the variance of a binomial distribution and the relationshi between the two distributions. A binomial distribution has mean n and variance n( ) n (n). Now, if n gets large and becomes small but n λ remains constant, the variance n (n) λ λ should tend toward λ. In fact, the Poisson distribution does have both its mean and its variance equal to λ. The mean of the Poisson distribution can easil be derived formall if one remembers a simle Talor series exansion of e x namel, Then, e x x x! 3 x 3!

59 59 E( Y ) λe λe λe λ The formal derivation of the fact that 0 ( ) λ e! λ e! λ λ λ λ ( )! 3 λ λ λ! 3! e λ V (Y ) is left as a challenge for the interested reader. [Hint: First, find E[Y(Y -)].] λ λ λ Examle 4.: The manager of an industrial lant is lanning to bu a new machine of either te A or te B. For each da s oeration, the number of reairs that machine A requires is a Poisson random variable with mean 0.0 t, where t denotes the time (in hours) of dail oeration. The number of dail reairs Y for machine B is a Poisson random variable with mean 0.t. The dail cost of oerating A is C A (t) 0t 40 ; for B, the cost is C B (t) 6t 40Y. Assume that the reairs take negligible time and that each night the machines are to be cleaned, so that the oerate like new machines at the start of each da. Which machine minimizes the exected dail cost if a Y consists of the following time sans?. 0 hours. 0 hours Solution: The exected cost for machine A is E [ C A ( t) ] 0t 40E( ) 0t 40[ V ( ) ( E( )) ] 0t 0.t 0.0t ] 4t 0.4t

60 60 Similarl,. Here, and E E [ C B ( t) ] 6t 40E( ) 6t 40[ V ( ) ( E( )) ] 6t 0.t 0.044t ] E 0.8t 0.576t [ (0)] 4(0) 0.4(0) 80 C A [ (0)] 0.8(0) 0.576(0) C B which results in the choice of machine B.. Here, E C A (0) 4(0) 0.4(0) and E C B (0) 0.8(0) 0.576(0) which results in the choice of machine A. [ ] 640 [ ] Histor and Alications Siméon-Denis Poisson (78-840), examiner and rofessor at the École Poltechnique of Paris for nearl fort ears, wrote over 300 aers in the fields of mathematics, hsics and astronom (see Figure 4.4). His most imortant works were a series of aers on definite integrals and his advances on Fourier series, roviding a foundation for later work in this area b Dirichlet and Riemann. However, it was his derivation of the exonential limit of the binomial distribution, much as we saw above, for which he is best known in robabilit and statistics. The derivation was given no secial emhasis, Cournot reublished it in 843 with calculations demonstrating the effectiveness of the aroximation. Although De Moivre had resented the exonential limit of the binomial distribution in the first edition of The Doctrine of Chances, ublished in 78, this distribution became known as the Poisson distribution. The Poisson distribution was rediscovered b von Bortkiewicz in 898. He tabulated the number of cavalrmen in the Prussian arm who died from a kick from a horse during a ear. To see how the Poisson distribution alies, first suose that a cavalrman can either be killed b a horsekick during a ear or not. Further suose that the chance of this rare event is the same for all soldiers and that soldiers have indeendent chances of being killed. Thus, the number of cavalrmen killed during a ear is a binomial random variable. However, the robabilit of being killed,, is ver small and the number of cavalrmen (trials) is ver large. Therefore, the Poisson limit is a reasonable model for the data. A comarison of the observed and theoretical relative frequencies is shown in Table 4.. Notice that the two agree well, indicating the Poisson is an adequate model for these data.

61 6 Figure 4.4. Siméon-Denis Poisson (78-840) Source: htt://en.wikiedia.org/wiki/simeon_poisson Table 4.. Deaths of Prussian Calvar Men Due to Kick b a Horse Number of Calvalrmen Killed During a Year Frequenc Relative Frequenc Theoretical Probabilit The Poisson distribution can be used as an aroximation for the binomial (large n and small ). When count data are observed, the Poisson model is often the first model considered. If the estimates of mean and variance differ significantl so that this roert of the Poisson is not reasonable for a articular alication, then one turns to other discrete distributions, such as the binomial (or negative binomial) for which the variance is less (greater) than the mean. The number of radioactive articles emitted in a given time eriod, number of telehone calls received in a given time eriod, number of equiment failures in a given time eriod, the number of defects in a secified length of wire, and the number of insects in a secified volume of soil are some of the man tes of data that have been modeled using the Poisson distribution. The Poisson Distribution λ λ ( ) e,! 0,,,.. for λ > 0 E (Y ) λ V (Y ) λ Exercises Let Y denote a random variable that has a Poisson distribution with mean λ 4. Find the following robabilities.

62 6 a. P(Y 5) b. P(Y < 5) c. P(Y 5) d. P(Y 5 Y ) The number of calls coming into a hotel s reservation center averages three er minute. a. Find the robabilit that no calls will arrive in a given one-minute eriod. b. Find the robabilit that at least two calls will arrive in a given one-minute eriod. c. Find the robabilit that at least two calls will arrive in a given two-minute eriod The Meteorolog Deartment of the Universit of Hawaii modeled the number of hurricanes coming within 50 nautical miles of Honolulu during a ear using a Poisson distribution with a mean of 0.45 (htt:// Using this model, determine the robabilities of the following events. a. At least one hurricane will come within 50 nautical miles of Honolulu during the next ear b. At most four hurricanes will come within 50 nautical miles of Honolulu during the next ear A certain te of coer wire has a mean number of.5 flaws er meter. a. Justif using the Poisson distribution as a model for the number of flaws in a certain length of this wire. b. Find the robabilit of having at least one flaw in a meter length of the coer wire. c. Find the robabilit of having at least one flaw in a 5-meter length of the coer wire Referring to Exercise 4.67, the cost of reairing the flaw in the coer wire is $8 er flaw. Find the mean and the standard deviation of the reair costs for a 0-meter length of wire in question Customer arrivals at a checkout counter in a deartment store have a Poisson distribution with an average of seven er hour. For a given hour, find the robabilities of the following events a. Exactl seven customers arrive b. No more than two customers arrive c. At least two customers arrive Referring to Exercise 4.69, if it takes aroximatel 0 minutes to service each customer, find the mean and the variance of the total service time connected to the customer arrivals for one hour. (Assume that an unlimited number of servers are available, so that no customer has to wait for service.) Is total service time highl likel to exceed 00 minutes? 4.7. Referring to Exercise 4.69, find the robabilities that exactl two customers will arrive in the follow two-hour eriods of time. a. Between :00.m. and 4:00.m. (one continuous two-hour eriod).

63 63 b. Between :00.m. and :00.m. and between 3:00.m. and 4:00.m.(two searate one-hour eriods for a total of two hours) The number of grasshoers er square meter of rangeland is often well modeled using the Poisson distribution. Suose the mean number of grasshoers in a secified region that has been grazed b cattle is 0.5 grasshoers er square meter. Find the robabilities of the following events. a. Five or more grasshoers in a randoml selected square meter in this region b. No grasshoers in a randoml selected square meter in this region c. At least one grasshoer in a randoml selected square meter in this region The number of articles emitted b a radioactive source is generall well modeled b the Poisson distribution. If the average number of articles emitted b the source in an hour is four, find the following robabilities. a. The number of emitted articles in a given hour is at least 6 b. The number of emitted articles in a given hour will be at most 3 c. No articles will be emitted in a given 4-hour eriod Chu (003) studied the number of goals scored during the 3 World Cu soccer games laed from 990 to 00. Onl goals scored during the 90 minutes of regulation la were considered. The average number of goals scored each game was.5. Assuming this mean continues to hold for other World Cu games, find the robabilities associated with the following events. a. At least six goals are scored during the 90 minutes of regulation la in a World Cu game b. No goals are scored during the 90 minutes of regulations la in a World Cu game When develoing a voice network, one imortant consideration is the availabilit of service. One measure of this is congestion. Congestion is the robabilit that the next call will be blocked. If there are c circuits, then congestion is the robabilit that c or more calls are in rogress. A large sales firm has an average of 8 calls during an minute of the business da. a. If the firm has 0 circuits, what is the robabilit that there will be congestion during an minute of the business da? b. The firm has decided to ugrade its sstem and wants to add enough circuits so that the robabilit of congestion is no more than % during an minute of the business da. How man circuits does the firm need? c. If the firm wants to add enough circuits that it could handle a 0% growth in telehone traffic and still have a robabilit of congestion of no more than % during an minute of the business da, how man circuits should it have? The number of fatalities due to shark attack during a ear is modeled using a Poisson distribution. The International Shark Attack File (ISAF) investigates sharkhuman interactions worldwide. Internationall, an average of 4.4 fatalities er ear occurred during 00 to 005. Assuming that this mean will remain constant for the next five ears (006 to 00), find the robabilities of the following events.

64 64 a. No shark fatalities will be recorded in a given ear. b. Sharks will cause at least 6 human deaths in a given ear c. No shark fatalities will occur in 006 to 00 d. At most shark fatalities will occur in 006 to Schmuland (00) exlored the use of the Poisson distribution to model the number of goals the hocke star, Wane Gretzk, scored during a game as an Edmonton Oiler. Gretzk laed 696 games with the following distribution of the number of goals scored: Points Number of Games a. Find the average number of goals scored er game. b. Using the average found in (a) and assuming that Gretzk s goals were scored according to a Poisson distribution, find the exected number of games in which 0,,,, 9 goals were scored. c. Comare the actual numbers of games with the exected numbers found in (b). Does the Poisson seem to be a reasonable model? The number of bacteria colonies of a certain te in samles of olluted water has a Poisson distribution with a mean of two er cubic centimeter. a. If four -cubic-centimeter samles of this water are indeendentl selected, find the robabilit that at least one samle will contain one or more bacteria columns. b. How man -cubic-centimeter samles should be selected to establish a robabilit of aroximatel 0.95 of containing at lest one bacteria colon? Let Y have a Poisson distribution with mean λ. Find E[Y(Y -)], and use the result to show that V(Y) λ A food manufacturer uses an extruder (a machine that roduces bite-size foods, like cookies and man snack foods) that has a revenue-roducing value to the firm of $300 er hour when it is in oeration. However, the extruder breaks down an average of twice ever 0 hours of oeration. If Y denotes the number of breakdowns during the time of oeration, the revenue generated b the machine is given b R 300t 75Y where t denotes hours of oerations. The extruder is shut down for routine maintenance on a regular schedule, and it oerates like a new machine after this maintenance. Find the otimal maintenance interval t 0 to maximize the exected revenue between shutdowns.

65 The Hergeometric Distribution 4.8. The Probabilit Function The distributions alread discussed in this chater have as their basic building block a series of indeendent Bernoulli trials. The examles, such as samling from large lots, deict situations in which the trials of the exeriment generate, for all ractical uroses, indeendent outcomes. But suose that we have a relativel small lot consisting of N items, of which k are defective. If two items are samled sequentiall, the outcomes for the second draw is significantl influenced b what haened on the first draw, rovided that the first item drawn remains out of the lot. A new distribution must be develoed to handle this situation involving deendent trials. In general, suose that a lot consists of N items, of which k are of one te (called successes) and N k are of another te (called failures). Suose that n items are samled randoml and sequentiall from the lot, and suose that none of the samled items is relaced. (This is called samling without relacement.) Let i if the ith draw results in a success, and let i 0 otherwise, where i,,, n. Let Y denote the total number of successes among the n samled items. To develo the robabilit distribution for Y, let us start b looking at a secial case for Y. One wa for successes to occur is to have,,,, 0,, n 0 We know that P, ) P( ) P( and this result can be extended to give Now, P( (,,, P( ) P( P( 0 n n, 0,, 0) ) P( 3, ) 0,, 0,,, ) P ) ( if the item is randoml selected; and similarl, k N n ) P ( ) k N because, at this oint, one of the k successes has been removed. Using this idea reeatedl, we see that

66 66 0), 0,,,,, ( n N n k N N k N N k N k N k P n rovided that k. A more comact wa to write the receding exression is to emlo factorials, arriving at the formula )! (! )! ( )! ( )! (! n N N n k N k N k k (The reader can check the equivalence of the two exressions.) An secified arrangement of successes and (n ) failures will have the same robabilit as the one just derived for all successes followed b all failures; the terms will merel be rearranged. Thus, to find P(Y ), we need onl count how man of these arrangements are ossible. Just as in the binomial case, the number of such arrangements is n. Hence, we have n N n k N k n N N n k N k N k k n Y P )! (! )! ( )! ( )! (! ) ( Of course, 0 k N and 0 n N. This formula is referred to as the hergeometric robabilit distribution. Notice that it arises from a situation quite similar to the binomial, excet that the trials here are deendent. Finding the robabilities associated with the hergeometric distribution can be comutationall intensive, eseciall as N and n increase. Calculators and comuter software are often valuable tools in determining hergeometric robabilities. Exlore the shae of the hergeometric distribution further using the alet. Examle 4.: Two ositions are oen in a coman. Ten men and five women have alied for a job at this coman, and all are equall qualified for either osition. The manager randoml hires eole from the alicant ool to fill the ositions. What is the robabilit that a man and a woman were chosen.

67 67 Solution: If the selections are made at random, and if Y denotes the number of men selected, then the hergeometric distribution would rovide a good model for the behavior of Y. Hence, 0 5 (0)(5) P ( Y ) () 5 5(4) Here, N 5, k, n, and Mean and Variance The techniques needed to derive the mean and the variance of the hergeometric distribution will be given in Chater 6. The results are V ( Y ) n E ( Y ) k N k n N k N n n N Because the robabilit of selecting a success on one draw is k/n, the mean of the hergeometric distribution has the same form as the mean of the binomial distribution. Likewise, the variance of the hergeometric matches the variance of the binomial, multilied b (N n)/(n ), a correction factor for deendent samles. Examle 4.3: In an assembl-line roduction of industrial robots, gear-box assemblies can be installed in one minute each if the holes have been roerl drilled in the boxes, and in 0 minutes each if the holes must be redrilled. Twent gear boxes are in stock, and two of these have imroerl drilled holes. Five gear boxes are selected from the twent available for installation in the next five robots in line.. Find the robabilit that all five gear boxes will fit roerl.. Find the exected value, the variance, and the standard deviation of the time it will take to install these five gear boxes. Solution: In this roblem, N 0; and the number of nonconforming boxes is k, according to the manufacturer s usual standards. Let Y denote the number of nonconforming boxes (that is, the number with imroerl drilled holes) in the samle of five. Then,

68 68 P( Y ) 0 5 ()(8568) 5, The total time T taken to install the boxes (in minutes) is T 0Y (5 Y ) 9Y 5 since each of Y nonconforming boxes takes 0 minutes to install, and the others take onl one minute. To find E(T) and V(T), we first need to calculate E(Y) and V(Y): and It follows that and k E ( Y ) n N 0 k ( ) k N n V Y n n N N 0 5 5(0.)( 0.) E( T ) 9E( Y ) 5 9(0.5) V ( T ) (9) V ( Y ) 8(0.355) Thus, installation time should average 9.5 minutes, with a standard deviation of minutes.

69 Histor and Alications Although the did not use the term hergeometric distribution, Bernoulli and de Moivre used the distribution to solve some of the robabilit roblems the encountered. In 899, Karl Pearson (see Figure 4.5) discussed using the hergeometrical series) to model data. But, it was not until 936 that the term hergeometric distribution actuall aeared in the literature (htt://members.aol.com/jeff570/h.html). Figure 4.5. Karl Pearson Source: htt://www-histor.mcs.st-andrews.ac.uk/pictdisla/pearson.html The rimar alication of the hergeometric distribution is in the stud of finite oulations. Although most oulations are finite, man are large enough so that the robabilities are relativel stable as units are drawn. However, as the fraction of the oulation samled becomes larger, the robabilities begin to change significantl with each new unit selected for the samle. The hergeometric distribution has been used extensivel in discrimination cases, qualit control, and surves.

70 70 The Hergeometric Distribution Exercises k N k n b P Y ( ), 0,,, k with 0 if a > b N a n k k E ( Y ) n and N k N n V ( Y ) n N n N 4.8. From a box containing five white and four red balls, two balls are selected at random, without relacement. Find the robabilities of the following events. a. Exactl one white ball is selected b. At least one white ball is selected c. Two white balls are selected, given that at lest one white ball is selected d. The second ball drawn is white 4.8. A coman has ten ersonal comuters (PCs) in its warehouse. Although all are new and still in boxes, four do not currentl function roerl. One of the coman s offices requests five PCs, and the warehouse foreman selects five from the stock of ten and shis to the requesting office. What is the robabilit that all five of the PC are not defective? Referring to Exercise 4.8, the office requesting the ersonal comuters returns the defective ones for reair. If it costs $80 to reair each PC, find the mean and the variance of the total reair cost. B Tchebsheff s Theorem, in what interval should ou exect, with a robabilit of at least 0.95, the reair costs of these five PCs to lie? A foreman has ten emloees, and the coman has just told him that he must terminate four of them. Three of the ten emloees belong to a minorit ethnic grou. The foreman selected all three minorit emloees (lus one other) to terminate. The minorit emloees then rotested to the union steward that the were discriminated against b the foreman. The foreman claimed that the selection had been comletel random. What do ou think? Justif our anwer A small firm has 5 emloees. Eight are single, and the other 7 are married. The owner is contemlating a change in insurance coverage, and she randoml selects five eole to get their oinions. a. What is the robabilit that she onl talks to single eole? b. What is the robabilit that she onl talks to married eole? c. What is the robabilit that single and 3 married eole are in the samle?

71 An auditor checking the accounting ractices of a firm samles four accounts from an accounts receivable list of. Find the robabilit that the auditor sees at least one ast-due account under the following conditions. a. There are two such accounts among the twelve. b. There are six such accounts among the twelve. c. There are eight such accounts among the twelve The ool of qualified jurors called for a high rofile case has whites, 9 blacks, 4 Hisanics, and Asians. From these, will be selected to serve on the jur. Assume that all qualified jurors meet the criteria for serving. Find the robabilities of the following events. a. No white is on the jur b. All nine blacks serve on the jur c. No Hisanics or Asians serve on the jur A student has a drawer with 0 AA batteries. However, the student does not realize that 3 of the batteries have lost their charge (will not work). She realizes that the batteries on her calculator are no longer working, and she is in a hurr to get to a test. She grabs two batteries from the drawer at random to relace the two batteries in her calculator. What is the robabilit that she will get two good batteries so that her calculator will work during the test? A cororation has a ool of six firms (four of which are local) from which the can urchase certain sulies. If three firms are randoml selected without relacement, find the robabilities of the following events a. All three selected firms are local b. At least one selected firm is not local Secifications call for a te of thermistor to test out at between 9000 and 000 ohms at 5 o C. Ten thermistors are available, and three of these are to be selected for use. Let Y denote the number among the three that do not conform to secifications. Find the robabilit distribution for Y (in tabular form) if the following conditions revail. a. The ten contain two thermistors not conforming to secifications b. The ten contain four thermistors not conforming to secifications A grou of 4 men and 6 women are faced with an extremel difficult and unleasant task, requiring 3 eole. The decide to draw straws to see who must do the job. Eight long and two short straws are made. A erson outside the grou is asked to hold the straws, and each erson in the grou selects one. Find the robabilit of the following events. a. Both short straws are drawn b men b. Both short straws are drawn b women c. The short straws are drawn b a man and a woman

72 An eight-clinder automobile engine has two misfiring sark lugs. If all four lugs are removed from one side of the engine, what is the robabilit that the two misfiring lugs are among them? Lot accetance samling rocedures for an electronics manufacturing firm call for samling n items from a lot of N items and acceting the lot if Y c, where Y is the number of nonconforming items in the samle. For an incoming lot of 0 transistors, 5 are to be samled. Find the robabilit of acceting the lot if c and the actual number of nonconforming transistors in the lot are as follows a. 0 b. c. d In the setting and terminolog of exercise 4.93, answer the same questions if c Two assembl lines (I and II) have the same rate of defectives in their roduction of voltage regulators. Five regulators are samled from each line and tested. Among the total of ten tested regulators, four are defective. Find the robabilit that exactl two of the defectives came from line I A 0-acre area has N raccoons. Ten of these raccoons were catured, marked so the could be recognized, and released. After five das, twent raccoons were catured. Let denote the number of those catured on the second occasion that was marked during the first samling occasion. Suose that catures at both time oints can be treated as random selections from the oulation and that the same N raccoons were in the area on both samling occasions (no additions or deletions). If N 30, what is the robabilit that no more than 5 of those catured during the second samling eriod were marked during the first samling occasion? If raccoons in the second samle were marked from having been caught in the first one, what value of N would result in the robabilit of this haing being the largest? 4.9 The Moment-generating Function We saw in earlier sections that, if g(y) is a function of a random variable Y, with robabilit distribution given b (), then [ g( Y )] E g( ) ( ) A secial function with man theoretical uses in robabilit theor is the exected value of e ty, for a random variable Y, and this exected value is called the momentgenerating function (mgf). The definition of a moment generating function is given in Definition 4.8.

73 73 Definition 4.8. The moment generating function (mgf) of a random variable is denoted b M(t) and defined to be ty M ( t) E( e ) The exected values of owers of a random variable are often called moments. Thus, E(Y) is the first moment of Y, and E(Y ) is the second moment of Y. One use for the moment-generating functions is that, in fact, it does generate moments of Y. When M(t) exists, it is differentiable in a neighborhood of the origin t 0, and the derivatives ma be taken inside the exectation. Thus () dm ( t) M ( t) dt d ty E( e ) dt d E e dt E Ye ty ty [ ] Now, if we set t 0, we have Going on to the second derivative, and In general, () M (0) E( Y ) ) M ( t) E( Y e ( ty () M (0) E( Y k ) M (0) E( Y ( k ) ) ) It often is easier to evaluate M(t) and its derivatives than to find the moments of the random variable directl. Other theoretical uses of the mgf will be discussed in later chaters. Examle 4.4: Evaluate the moment generating function for the geometric distribution, and use it to find the mean and the variance of this distribution. Solution: For the geometric variable Y, we have

74 74 M ( t) E( e 0 e ty t ) q t ( qe ) 0 t t [ qe ( qe ) ] qe t qe because the series is geometric with a common ratio of qe t. Note: For the series to converge and the moment generating function to exist, we must have qe t <, which is the case if t < ln(/q). It is imortant for the moment generating function to exist for t in a neighborhood about 0, and it does here. To evaluate the mean, we have t () 0 qe M ( t) t ( qe ) t qe t ( qe ) and () q q M ( t) ( q) To evaluate the variance, we first need t Now, and Hence, M () ( t) () ( Y ) M ( t) E t t t t ( qe ) qe qe ()( qe ) t 4 ( qe ) t qe q e M t ( qe ) 3 t ( ) q q q( q) (0) 3 ( q) ( ) qe t

75 75 V ( Y ) E Y ( ) [ E( Y )] q( q) q q Moment-generating functions have imortant roerties that make them extremel useful in finding exected values and in determining the robabilit distributions of random variables. These roerties will be discussed in detail in Chaters 5 and 6, but one such roert is given in Exercise The Probabilit-generating Function In an imortant class of discrete random variables, Y takes integral values (Y 0,,, 3, ) and, consequentl, reresents a count. The binomial, geometric, hergeometric, and Poisson random variables all fall in this class. The following examles resent ractical situations involving integral-valued random variables. One, tied to the theor of queues (waiting lines), is concerned with the number of ersons (or objects) awaiting service at a articular oint in time. Understanding the behavior of this random variable is imortant in designing manufacturing lants where roduction consists of a sequence of oerations, each of which requires a different length of time to comlete. An insufficient number of service stations for a articular roduction oeration can result in a bottleneck the forming of a queue of roducts waiting to be serviced which slows down the entire manufacturing oeration. Queuing theor is also imortant in determining the number of checkout counters needed for a suermarket and in designing hositals and clinics. Integer-valued random variables are extremel imortant in studies of oulation growth, too. For examle, eidemiologists are interested in the growth of bacterial oulations and also in the growth of the number of ersons afflicted b a articular disease. The number of elements in each of these oulations is an integral-valued random variable. A mathematical device that is ver useful in finding the robabilit distributions and other roerties of integral-valued random variables in the robabilit-generating function P(t), which is defined in Definition 4.9. Definition 4.9. The robabilit generating function of a random variable is denoted b P(t) and is defined to be Y P ( t) E( t ) If Y is an integer-valued random variable, with

76 76 Then ( Y ), 0,,, P i Y P( t) E( t ) t t 0 The reason for calling P(t) a robabilit generating function is clear when we comare P(t) with the moment-generating function M(t). Particularl, the coefficient of t i in P(t) is the robabilit i. If we know P(t) and can exand it into a series, we can determine () as the coefficient of t. Reeated differentiation of P(t) ields factorial moments for the random variable Y Definition 4.7: The kth factorial moment for a random variable Y is defined to be μ where k is a ositive integer. E[ Y ( Y )( Y ) ( Y k [ k ] )] When a robabilit-generating function exists, it can be differentiated in a neighborhood of t. Thus, with Y P ( t) E( t ) we have () dp( t) P ( t) dt d Y E( t ) dt d Y E t dt Setting t, we have Similarl, and In general, and P P E Yt Y [ ] () P () E( Y ) ( t) E[ Y ( Y ) t ( ) Y P () () E[ Y ( Y )] ( t) E[ Y ( Y ) ( Y k ) t ( k ) Y k P ( k ) () E[ Y ( Y ) ( Y k )] μ [ k ] ] ]

77 77 Examle 4.5: Find the robabilit-generating function for the geometric random variable, and use this function to find the mean. Solution: P( t) E Y ( t ) 0 t 0 qt where qt < for the series to converge. Now, Setting t, P () ( t) d dt qt () q P ( ) q ( qt) ( q) which is the mean of a geometric random variable. q ( qt) q Since we alread have the moment-generating function to assist us in finding the moment of a random variable, we might ask how knowing P(t) can hel us. The answer is that in some instances it ma be exceedingl difficult to find M(t) but eas to find P(t). Alternativel, P(t) ma be easier to work with in a articular setting. Thus, P(t) siml rovides an additional tool for finding the moments of a random variable. It ma or ma not be useful in a given situation. Finding the moments of a random variable is not the major use of the robabilitgenerating function. Its rimar alication is in deriving the robabilit function (and hence the robabilit distribution) for related integral-valued random variables. For these alications, see Feller (968), Parzen (964), and Section 7.7. Exercises Find the moment-generating function for the Bernoulli random variable Derive the mean and variance of the Bernoulli random variable using the moment generating function derived in Exercise 4.97.

78 Show that the moment-generating function for the binomial random variable is given b t ( e q) n M ( t) Derive the mean and variance of the binomial random variable using the momentgenerating function derived in Exercise Show that the moment-generating function for the Poisson random variable with mean λ is given b t λ ( e ) M ( t) e 4.0. Derive the mean and variance of the Poisson random variable using the momentgenerating function derived in Exercise Show that the moment-generating function for the negative binomial random variable is given b r M ( t) t qe Derive the mean and variance of the negative binomial random variable using the moment-generating function derived in Exercise Derive the robabilit-generating function of a Poisson random variable with arameter λ Using the robabilit-generating function derived in Exercise 4.05 find the first and second factorial moments of a Poisson random variable. From the first two factorial moments, find the mean and variance of the Poisson random variable Derive the robabilit-generating function of the binomial random variable of n trials, with robabilit of success Find the first and second factorial moments of the binomial random variable in Exercise Using the first two factorial moments, find the mean and variance of the Poisson random If is a random variable with moment-generating function M(t), and Y is a function of given b Y a b, show that the moment-generating function for Y is e tb M(at) Use the result of Exercise 4.09 to show that E ( Y ) ae( ) b and V ( Y ) a V ( )

79 79 4. Markov Chains Consider a sstem that can be in an of a finite number of states. Assume that the sstem moves from state to state according to some rescribed robabilit law. The sstem, for examle, could record weather conditions from da to da, with the ossible states being clear, artl cloud, and cloud. Observing conditions over a long eriod of time would allow one to find the robabilit of its being clear tomorrow given that it is artl cloud toda. Let i denote the state of the sstem at time oint i, and let the ossible states be denoted b S,, S m, for a finite integer m. We are interested not in the elased time between transitions from one time state to another, but onl in the states and the robabilities of going from one state to another that is, in the transition robabilities. We assume that P S ) ( i S k i j jk where jk is the transition robabilit from S j to S k ; and this robabilit is indeendent of i. Thus, the transition robabilities deend not on the time oints, but onl on the states. The event ( i S k i- S j ) is assumed to be indeendent of the ast histor of the rocess. Such a rocess is called a Markov chain with stationar transition robabilities. The transition robabilities can convenientl be dislaed in a matrix: P m m m m mm Let 0 denote the starting state of the sstem, with robabilities given b (0) k P( 0 S k P ) and let the robabilit of being in state S k after n stes be given b robabilities are convenientl dislaed b vectors: (n) k. These and (0) [ [ (0),, (0),, (0) m,, ( n) ( n) ( n) ( n) m ] ] To see how (0) and () are related, consider a Markov chain with onl two states, so that

80 80 P There are two was to get to state after one ste: either the chain starts in state and stas there, or the chain starts in state and then moves to state in one ste. Thus, Similarl, In terms of matrix multilication, and, in general, () (0) (0) () (0) (0) () (0) P ( n) ( n) P P is said to be regular if some ower of P (P n for some n) has all ositive entries. Thus, one can get from state S j to state S k, eventuall, for an air (j, k). (Notice that the condition of regularit rules out certain chains that eriodicall return to certain states.) If P is regular, the chain has a stationar (or equilibrium) distribution that gives the robabilities of its being in the resective states after man transitions have evolved. In other words, must have a limit π j, as n. Suose that such limits exist; then (n) j π π,, π ) must satisf ( n π πp because ( n) ( n) P will have the same limit as ( n). Examle 4.6: A suermarket stocks three brands of coffee A, B, and C and customers switch from brand to brand according to the transition matrix 3/ 4 P 0 / 4 / 4 / 3 / 4 0 / 3 / where S corresonds to a urchase of brand A, S to brand B, and S 3 to brand C; that is, ¾ of the customers buing brand A also bu brand A the next time the urchase coffee, whereas ¼ of these customers switch to brand B.. Find the robabilit that a customer who bus brand A toda will again urchase brand A two weeks from toda, assuming that he or she urchases coffee once a week. In the long run, what fractions of customers urchase the resective brands?

81 8 Solution:. Assuming that the customer is chosen at random, his or her transition robabilities are given b P. The given information indicates that (0) (, 0, 0); that is, the customer starts with a urchase of brand A. Then () (0) 3 P,, gives the robabilities for the next week s urchase. The robabilities for two weeks from now are given b () () P That is, the chance of the customer s urchasing A two weeks from now is onl 9/6.. The answer to the long-run frequenc ratio is given b π, the stationar distribution. The equation π πp ields the sstem of equations 3 π π π π π π π π 3 π π , 48, Combining these equations with the fact that π π π ields 3 π,, Thus, the store should stock more brand B coffee than either brand A or brand C. Examle 4.7: Markov chains are used in the stud of robabilities connected to genetic models. Recall from Section 3. that genes come in airs. For an trait governed b a air of genes, an individual ma have genes that are homozgous dominant (GG), heterozgous (Gg), or homozgous recessive (gg). Each offsring inherits one gene of a air from each arent, at random and indeendentl. Suose that an individual of unknown genetic makeu is mated with a heterozgous individual. Set u a transition matrix to describe the ossible states of a resulting offsring and their robabilities. What will haen to the genetic makeu of the offsring after man generations of mating with a heterozgous individual?

82 8 Solution: If the unknown is homozgous dominant (GG) and is mated with a heterozgous individual (Gg), the offsring has a robabilit of ½ of being homozgous dominant and a robabilit of ½ of being heterozgous. If two heterozgous individuals are mated, the offsring ma be homozgous dominant, heterozgous, or homozgous recessive with robabilities ¼, ½, and ¼, resectivel. If the unknown is homozgous recessive (gg), the offsring of it and a heterozgous individual has a robabilit of ½ of being homozgous recessive and a robabilit of ½ of being heterozgous. Following along these lines, a transition matrix from the unknown arent to an offsring is given b r h d r h d P The matrix P has all ositive entries (P is regular); and hence a stationar distribution exists. From the matrix equation π πp we obtain π π π π π π π π π π Since 3 π π π, the second equation ields π. It is then eas to establish that 4 3 π π. Thus, 4,, 4 π No matter what the genetic makeu of the unknown arent haened to be, the ratio of homozgous dominant to heterozgous to homozgous recessive offsring among its descendants, after man generations of mating with heterozgous individuals, should be ::.

83 83 An interesting examle of a transition matrix that is not regular is formed b a Markov chain with absorbing states. A state S i is said to be absorbing if ii and ij 0 for j i. That is, once the sstem is in state S i, it cannot leave it. The transition matrix for such a chain can alwas be arranged in a standard form, with the absorbing states listed first. For examle, suose that a chain has five states, of which two are obsorbing. Then P can be written as I 0 P R Q where I is a x identit matrix and 0 is a matrix of zeros. Such a transition matrix is not regular. Man interesting roerties of these chains can be exressed in terms of R and Q (see Kemen and Snell, 983). The following discussion will be restricted to the case in which R and Q are such that it is ossible to get to an absorbing state from ever other state, eventuall. In that case, the Markov chain eventuall will end u in an absorbing state. Questions of interest then involve the exected number of stes to absortion and the robabilit of absortion in the various absorbing states. Let m ij denote the exected (or mean) number of times the sstem is in state S j, given that it started in S i, for nonabsorbing states S i and S j. From S i, the sstem could go to an absorbing state in one ste, or it could go to a nonabsorbing state sa S k and eventuall be absorbed from there. Thus, m ij must satisf m ij ij k ik m kj Where the summation is over all nonabsorbing states and, if i j 0 otherwise ij, The term ij accounts for the fact that, if the chain goes to an absorbing state in one ste, it was in state S i one time. If we denote the matrix of m ij terms b M, the receding equation can then be generalized to M I QM or M ( I Q)

84 84 Matrix oerations, such as inversion, will not be discussed here. The equations can be solved directl if matrix oerations are unfamiliar to the reader.) The exected number of stes to absortion, from the nonabsorbing starting state S i, will be denoted b m i and given siml b m i m ik k again summing over nonabsorbing states. Turning now to the robabilit of absortion into the various absorbing states, we let a ij denote the robabilit of the sstem s being absorbed in state S j, given that it started in state S i, for nonabsorbing S i and absorbing S j. Reeating the receding argument, the sstem could move to S j in one ste, or it could move to a nonabsorbing state S k and be absorbed from there. Thus, a ij satisfies a ij ij k ik a kj where the summation occurs over the nonabsorbing states. If we denote the matrix of a ij terms b A, the receding equation then generalizes to or A R QA A (I Q) MR R The following examle illustrates the comutations. Examle 4.8: A manager of one section of a lant has different emloees working at level I and at level II. New emloees ma enter his section at either level. At the end of each ear, the erformance of each emloee is evaluated; emloees can be reassigned to their level I or II jobs, terminated, or romoted to level III, in which case the never go back to I or II. The manager can kee track of emloee movement as a Markov chain. The absorbing states are termination (S ) and emloment at level III (S ); the nonabsorbing states are emloment at level (S 3 ) and emloment at level II (S 4 ). Records over a long eriod of time indicate that the following is a reasonable assignment of robabilities:

85 85 0 P Thus, if an emloee enters at a level I job, the robabilit is 0.5 that she will jum to level II work at the end of the ear, but the robabilit is 0. that she will be terminated.. Find the exected number of evaluations an emloee must go through in this section.. Find the robabilities of being terminated or romoted to level III eventuall. Solution:. For the P matrix, and Thus, and It follows that M ( I Q) 0. R Q I Q m 3 and m 6 7 m 8 m m m In other words, a new emloee in this section can exect to remain there through 0/7 evaluation eriods if she enters at level I, whereas she can exect to remain there through 8/7 evaluations if she enters at level II.. The fact that a3 a3 A MR a4 a4 imlies that an emloee entering at level I has a robabilit of 4/7 of reaching level III, whereas an emloee entering at level II has a robabilit of 5/7 of reaching level III. The robabilities of termination at levels I and II are therefore 3/7 and /7, resectivel.

86 86 Examle 4.9: Continuing the genetics roblem in Examle 4.7, suose that an individual of unknown genetic makeu is mated with a known homozgous dominant (GG) individual. The matrix of transition robabilities for the first-generation offsring then becomes r h d r h d P which has one absorbing state. Find the mean number of generations until all offsring become dominant. Solution: In the notation used earlier, 0 0 Q 0 Q I and M Q I 0 ) ( Thus, if the unknown is heterozgous, we should exect all offsring to be dominant after two generations. If the unknown is homozgous recessive, we should exect all offsring to be dominant after three generations. Notice that 0 0 MR A which siml indicates that we are guaranteed to reach the full dominant state eventuall, no matter what the genetic makeu of the unknown arent ma be. Exercises 4.. A certain cit rides itself on having sunn das. If it rains one da, there is a 90% chance that it will be sunn the next da. If it is sunn one da, there is a 30% chance that it will rain the following da. (Assume that there are onl sunn or rain

87 87 das.) Does the cit have sunn das most of the time? In the long run, what fraction of all das are sunn? 4.. Suose a car rental agenc in a large cit has three locations: downtown location (labeled A), airort location (labeled B), and a hotel location (labeled C). The agenc has a grou of deliver drivers to serve all three locations. Of the calls to the Downtown location, 30% are delivered in downtown area, 30% are delivered to the airort, and 40% are delivered to the hotel. Of the calls to the airort location, 40% are delivered in downtown area, 40% are delivered to the airort, and 0% are delivered to the hotel. Of the calls to the hotel location, 50% are delivered in the downtown area, 30% are delivered to the airort area, and 0% are delivered to the hotel area. After making a deliver, a driver goes to the nearest location to make the next deliver. This wa, the location of a secific driver is determined onl b his or her revious location. a. Give the transition matrix. b. Find the robabilit that a driver who begins in the downtown location will be at the hotel after two deliveries. c. In the long run, what fraction of the total number of stos does a driver make at each of the three locations? 4.3. For a Markov chain, show that P (n) P (n - ) P 4.4. Suose that a article moves in unit stes along a straight line. At each ste, the article either remains where it is, moves one ste to the right, or moves one ste to the left. The line along which the article moves has barriers at 0 and at b, a ositive integer, and the article onl moves between these barriers; it is absorbed if it lands on either barrier. Now, suose that the article moves to the right with robabilit and to the left with robabilit q. a. Set u the general form of the transition matrix for a article in this sstem. b. For the case b 3 and q, show that the absortion robabilities are as follows: a 0 q q 3 q 3 c. In general, it can be shown that a 0 q q 3 q 3

88 88 a 0 q q, b q b q B taking the limit of a j0 as ½, show that b j a j 0, q b d. For the case b 3, find an exression for the mean time to absortion from state j, with q. Can ou generalize this result? 4.5. Suose that n white balls and n black balls are laced in two urns so that each urn contains n balls. A ball is randoml selected from each urn and laced in the oosite urn. (This is one ossible model for the diffusion of gases.) a. Number the urns and. The state of the sstem is the number of black balls in urn. Show that the transition robabilities are given b the following quantities: j jj, j > n 0 jj j( n n j) n j jj, j < n n jk 0, otherwise b. After man transitions, show that the stationar distribution is satisfied b n j j n n Give an intuitive argument as to wh this looks like a reasonable answer Suose that two friends, A and B, toss a balanced coin. If the coin comes u heads, A wins $ from B. If it comes u tails, B wins $ from A. The game ends onl

89 89 when one laer has all the other s mone. If A starts with $ and B with $3, find the exected duration of the game and the robabilit that A will win. 4.. Activities for Students: Simulation Comuters lend themselves nicel to use in the area of robabilit. Not onl can comuters be used to calculate robabilities, but the can also be used to simulate random ariables from secified robabilit distributions. A simulation, erformed on the comuter, ermits the user to analze both theoretical and alied roblems. A simujlated model attemts to cou the behavior of a situation under consideration; ractical alications include models of inventor control roblems, queuing sstems, roduction lines, medical sstems, and flight atterns of major jets. Simulation can also be used to determine the behavior of a comlicated random variable whose recise robabilit distribution function is difficult to evaluate mathematicall. Generating observations from a robabilit distribution is based on random numbers on [0, ]. A random number R i on the interval [0, ] is chosen, on the condition that each number between 0 and has the same robabilit of being selected. A sequence of numbers that aears to follow a certain attern or trend should not be considered random. Most comuter languages have built-in random generators that give a number on [0, ]. If a built-in generator is not available, algorithms are available for setting u one. One basic technique used is a congruential method, such as a linear congruential generator (LCG). The generator most commonl used is x i ( ax c)(mod m), i, i, Where x i, a, c, and m are integers and 0 x i <m. If c 0, this method is called a multilicative-congruential sequence. The values of x i generated b the LCG are contained in the interval [0, m). To obtain a value in [0, ), we evaluate x i /m. A random number generator should have certain characteristics, such as the following:. The numbers roduced (x i /m) should aear to be distributed uniforml on [0, ]; that is, the robabilit function for the numbers should be constant over the interaval [0, ]. The numbers should also be indeendent of each other. An statistical tests can be alied to check for uniformit and indeendence, such as comaring the number of occurrences of a digit in a sequence with the exected number of occurrences for that digit.. The random number generator should have a long, full eriod; that is, the sequence of numbers should not begin reeating or ccling too quickl. A sequence has a full eriod () if m. It has been shown that an LCG has a full eriod if and onl if the following are true (Kenned and Gentle 980,. 37). a. The onl ositive integer that (exactl) divides both m and c is. b. If q is a rime number that divides m, then q divides a. c. If 4 divides m, then 4 divides a.

90 90 This imlies that, if c is odd and a is divisible b 4, when a full-eriod generator is used, x 0 can be an integer between 0 and m without affecting the generator s eriod. A good choice for m is b, where b is the number of bits. Based on the just-mentioned relationshis of a, c, and m, the following values have been found to be satisfactor for use on a microcomuter: a 5, 73; c 3,849; m 6 65,536 (Yang and Robinson 986,. 6). 3. It is beneficial to be able to reuse the same random numbers in a different simulation run. The receding LCG has this abilit, because all values of x i are determined b the initial value (seed) x 0. Because of this, the comuted random numbers are referred to as seudo-random numbers. Even though these numbers are not trul random, careful selection of a, c, m and x 0 will ield values for x i that behave as random numbers and ass the aroriate statistical tests, as described in characteristic. 4. The generator should be efficient that is, fast and in need of little storage. Given that a random number R i on [0, ] can be generated, we will now consider a brief descrition of generating discrete random variables for the distributions discussed in this chater. 4.. Bernoulli Distribution Let reresent the robabilit of success. If R i, then i ; otherwise, i Binomial Distribution A binomial random variable i can be exressed as the sum of indeendent Bernoulli random variables Y j ; that is i Y j, where j,,, n. Thus to simulate i witn arameters n and, we simulate n Bernoulli random variables, as stated reviousl, i is equal to the sum of the n Bernoulli variables Geometric Distribution Let i reresent the number of failures rior to the first success, with being robabilit of success, i m, where m is the number of R i s generated rior to generating an R i such that R i Negative Binomial Distribution Let i reresent the number of trials rior to the rth success, with being the robabilit of success. A negative binomial random variable i can be exressed as the sum of r indeendent geometric random variables Y j ; that is, i Y j, where j,,, r. Thus, to simulate i with arameter, we simulate r geometric random variables, as stated reviousl. i is equal to the sum of the r geometric random variables Poisson Distribution Generating Poisson random variables will be discussed at the end of Chater 5.

91 9 Let us consider some simle examles of ossible uses for simulating discrete random variables. Suose that n items are to be insected from on roduction line and that n items are to be insected from another. Let reresent the robabilit of a defective from line, and let reresent the robabilit of a defective from line. Let be a binomial random variable with arameters n and. Let Y be a binomial random variable with arameters n and. A variable of interest is W, which reresents the total of defective items observed in both roduction lines. Let W Y. Unless, the distribution of W will not be binomial. To see how the distribution of W will behave, we can erform a simulation. Useful information could be obtained from the simulation b looking at a histogram of the values of W i generated and considering the values and Y; is binomial with n 7, 0.; and Y is binomial with n 8, 0.6. Defining W Y, a simulation roduced the histogram shown in Simulation. The samle mean was 6., with a samle standard deviation of.76. In Chater 6, we will be able to show that these values are ver close to the exected values of μ W 6. and σ W.74. (This calculation will make sense after we discuss the linear function of random variables.) From the histogram, we see that the robabilit that the total number of defective items is at least 9 is given b Another examle of interest might be the couon-collector roblem, which incororates the geometric distribution. Suose that there are n distinct colors of couons. We assume that, each time someone obtains a couon, it is equall likel to be an one of the n colors and the selection of the couon is indeendent of an reviousl obtained couon. Suose that an individual can redeem a set of couons for a rize if each ossible color couon is reresented in the set. We define the random variable as reresenting the total number of couons that must be selected to comlete a set of each color couon at random. Questions of interest might include the following:. What is the exected number of couons needed in order to obtain this comlete set; that is, what is E()?. What is the standard deviation of? 3. What is the robabilit that one must select at most x couons to obtain this comlete set? Instead of answering these questions b deriving the distribution function of, one might tr simulation. Two simulations (Simulations and 3) follow. The first histogram reresents a simulation where n, the number of different color couons, is equal to 5. The samle mean in this case was comuted to be.06, with a samle standard deviation of Suose that one is interested in finding P( 0). Using the results of the simulations, the relatie frequenc robabilit is given as It might also be noted that, from this simulation, the largest number of couons needed to obtain the comlete set was Binomial Proortions Count data are often more convenientl discussed as a roortion as in the roortion of insected cars that have oint defects, or the roortion of samled voters who favor a

92 9 certain candidate. For a binomial random variable on n trials, and a robabilit of success on an one trial, the roortion of successes is /n. Generate 00 values of /n and lot them on a line lot or histogram for each of the following cases:. n 0, 0.. n 40, n 0, n 40, 0.5 Comare the four lots, and discuss an atterns ou observe in their centering, variation, and smmetr. Can ou roose a formula for the variance of /n? Simulation Simulation

93 93 Simulation Waiting for Blood A local blood bank knows that about 40% of its donors have A blood. It needs k 3 A donors toda. Generate a distribution of values for, the number of donors that must be

94 94 tested sequentiall to find the three A donors needed. What are the aroximate exected value and standard deviation of from our data? Do these values agree with the theor? What is the estimated robabilit that the blood bank must test 0 or more eole to find the three A donors? How will the answers to these questions change if k 4? 4.3. Summar The outcomes of interest in most investigations involving random events are numerical. The simlest numbered outcomes to model are counts, such as the number of nonconforming arts in a shiment, the number of sunn das in a month, or the number of water samles that contain a ollutant. One amazing result of robabilit theor is the fact that a small number of theoretical distributions can cover a wide arra of alications. Six of the most useful discrete robabilit distributions are introduced in this chater. The Bernoulli random variable is siml an indicator random variable; it uses a numerical code to indicate the resence or absence of a characteristic. The binomial random variable counts the number of successes among a fixed number n of indeendent events, each with the same robabilit of success. The geometric random variable counts the number of failures one obtains rior needs to conduct sequentiall until the first success is seen when conducting indeendent Bernoulli trials, each with the robabilit of success. The negative binomial random variable counts the number of failures observed when Bernoulli trials, each with the robabilit of success, are conducted sequentiallt until the rth success is obtained. The negative binomial ma be derived from several other models and is often considered as a model for discrete count data if the variance exceeds the mean. The Poisson random variable arises from counts in a restricted domain of time, area, or volume, and is most useful for counting fairl rare outcomes. The hergeometric random variable counts the number of successes in samling from a finite oulation, which makes the sequential selections deendent on one another. These theoretical distributions serve as models for read daa that might arise in our quest to imrove a rocess. Each involves assumtions that should be checked carefull before the distribution is alied. Sulementar Exercises 4.7. Construct robabilit histograms for the binomial robabilit distribution for n 7, and 0., 0.5, and 0.8. Notice the smmetr for 0.5 and the direction of skewness for 0. and Construct a robabilit histogram for the binomial robabilit distribution for n 0 and 0.5. Notice that almost all of the robabilit falls in the interval 5 5.

95 The robabilit that a single field radar set will detect an enem lane is 0.8. Assume that the sets oerate indeendentl of each other. a. If we have five radar sets, what is the robabilit that exactl four sets will detect the lane? b. At least one set will detect the lane? 4.0. Suose that the four engines of a commercial aircraft were arranged to oerate indeendentl and that the robabilit of in-flight failure of a single engine is 0.0. What are the robabilities that, on a given flight, the following events occur? a. No failures are observed b. No more than one failure is observed c. All engines fail 4.. Samling for defectives from among large lots of a manufactured roduct ields a number of defectives Y that follows a binomial robabilit distribution. A samling lan consists of secifing the number n of items to be included in a samle and an accetance number a. The lot is acceted if Y a, and is rejected if Y > a. Let denote the roortion of defectives in the lot. For n 5 and a 0, calculate the robabilit of lot accetance if the following lot roortions of defectives exist. a. 0 b. 0.3 c..0 d. 0. e. 0.5 A grah showing the robabilit of lot accetance as a function of lot fraction defective is called the oerating characteristic curve for the samle lan. Construct this curve for the lan n 5, a 0. Notice that a samling lan is an examle of statistical inference. Acceting or rejecting a lot based on information contained in the samle is equivalent to concluding that the lot is either good or bad, resectivel. Good imlies that a low fraction of items are defective and, therefore, that the lot is suitable for shiment. 4.. Refer to exercise 4.. Construct the oerating characteristic curve for a samling lan with the following values a. n 0, a 0 b. n 0, a c. n 0, a For each, calculate P(lot accetance) for 0.05, 0., 0.3, 0.5, and.0. Our intuition suggests that samling lan (a) would be much less likel to accet bad lots than would lan (b) or lan (c). A visual comarison of the oerating characteristic curves will confirm this suosition Refer to exercise 4.. A qualit control engineer wishes to stud two alternative samle lans: n 5, a ; and n 5, a 5. On a sheet of grah aer, construct the oerating characteristic curves for both lans; make use of accetance robabilities at 0.05, 0.0, 0.0, 0.30, and 0.40 in each case.

96 96 a. If ou were a seller roducing lots whose fraction of defective items ranged from 0 to 0.0, which of the two samling lans would ou refer? b. If ou were a buer wishing to be rotected against acceting lots with a fraction defective exceed 0.30, which of the two samling lans would ou refer? 4.4. At an archaeological site that was an ancient swam, the bones from Trannosaurus rex skeletons have been unearthed. The bones do not show an sign of disease or malformation. It is thought that these animals wandered into a dee area of the swam and became traed in the swam bottom. The right femur bones (thigh bones) were located and 5 of these right femurs are to be randoml selected without relacement for DNA testing to determine gender. Let be the number out of the 5 selected right femurs that are from males. a. Based on how these bones were samled, exlain wh the robabilit distribution of is not binomial. b. Would the binomial distribution rovide a good aroximation to the robabilit distribution of? Justif our answer. c. Suose that the grou of Trannasaurus whose remains were found in the swam had been made u of males and females. What is the robabilit that all 5 in the samle to be tested are male? (Do not use the binomial aroximation.) d. The DNA testing revealed that all 5 femurs tested were from males. Based on this result and our answer from art (b), do ou think that males and females were equall reresented in the grou of brontosaurs stuck in the swam? Exlain For a certain section of a ine forest, the number Y of diseased trees er acre has a Poisson distribution with mean λ 0. To treat the trees, sraing equiment must be rented for $50. The diseased trees are sraed with an insecticide at a cost of $5 er tree. Let C denote the total sraing cost for a randoml selected acre. a. Find the exected value and the standard deviation for C. b. Within what interval would ou exect C to lie with a robabilit of at least 0.80? 4.6. In checking river water samles for bacteria, a researcher laces water in a culture medium to grow colonies of certain bacteria, if resent. The number of colonies er dish averages fifteen for water samles from a certain river. a. Find the robabilit that the next dish observed will have at least ten colonies. b. Find the mean and the standard deviation of the number of colonies er dish. c. Without calculating exact Poisson robabilities, find an interval in which at least 75%of the colon count measurements should lie The number of vehicles assing a secified oint on a highwa averages eight er minute. a. Find the robabilit that at least 5 vehicles will ass this oint in the next minute. b. Find the robabilit that at least 5 vehicles will ass this oint in the next two minutes. c. What assumtions must ou make for ou answers in (a) and (b) to be valid?

97 A roduction line often roduces a variable number N of items each da. Suose that each item roduced has the same robabilit of not conforming to manufacturing standards. If N has a Poisson distribution with mean λ, then the number of nonconforming items in one da s roduction Y has a Poisson distribution with mean λ. The average number of resistors roduced b a facilit in one da has a Poisson distribution, with a mean of 00. Ticall, 5% of the resistors roduced do not meet secifications. a. Find the exected number of resistors that will not meet secifications on a given da. b. Find the robabilit that all resistors will meet the secifications on a given da. c. Find the robabilit that more than five resistors will fail to meet secifications on a given da The mean number of customers arriving in a bank during a randoml selected hour is 4. The bank manager is considering reducing the number of tellers, but she wants to be sure that lines do not get too long. She decides that, if no more than two customers come in during a 5-minute eriod, two tellers (instead of the current 3) will be sufficient. a. What is the robabilit no more than customers will come in the bank during a randoml selected 5-minute eriod? b. What is the robabilit that more than two customers will come in during two consecutive 5-minute eriods? c. The manager records the number of customers coming in during 5-minute time eriods until she observes a time eriod during which more than two customers arrive. Eight time eriods have been recorded, each with two or fewer customers arriving in each. What is the robabilit that more than 4 time eriods will be observed before having more than two customers arriving during a timer 5-minute eriod An interviewer is given a list of otential eole she can interview. Suose that the interviewer needs to interview 5 eole and that each erson indeendentl agrees to be interviewed with robabilit 0.6. Let be the number of eole who refuse the interview before she obtains her necessar number of interviews. a. What is the robabilit that no more than two eole will refuse the interview before she finds 5 eole to interview? b. What is the exected value and variance of the number of eole who refuse the interview before obtaining 5 eole to interview? 4.3. Three men fli coins to see who as for coffee. If all three match (all heads or all tails), the fli again. Otherwise, the odd man as for coffee. a. What is the robabilit that the will need to fli the coins more than once? b. What is the robabilit the will need to toss the coins more than three times? c. Suose the men have flied the coins three times and all three times the matched. What is the robabilit that the will need to fli the coins more than three more times (more than 6 times total)? 4.3. A certain te of bacteria cell divides at a constant rate λ over time. Thus, the robabilit that a articular cell will divide in a small interval of time t is aroximatel λt. Given that a oulation starts out at time zero with k cells of this te, and cell

98 98 divisions are indeendent of one another, the size of the oulation at time t, Y(t), has the robabilit distribution n λkt λt nk P[ Y ( t) n] e ( e ), k n k, k, a. Find the exected value of Y(t) in terms of λ and t. b. If, for a certain te of bacteria cell, λ 0. er second, and the oulation starts out with two cells at time zero, find the exected oulation size after 5 seconds In a certain region, the robabilit of at least one child contracting malaria during a given week is 0.. Find the average number of cases during a four-week eriod, assuming that a erson contracting malaria is indeendent of another erson contracting malaria The robabilit that an one vehicle will turn left at a articular intersection is 0.. The left-turn lane at this intersection has room for three vehicles. If five vehicles arrive at this intersection while the light is red, find the robabilit that the left-turn lane will hold all of the vehicles that want to turn left Referring to Exercise 4.34, find the robabilit that six cars must arrive at the intersection while the light is red to fill u the left-turn lane For an robabilit (), ( ) if the sum is taken over all ossible values that the random variable in question can assume. Show that this is true for the following distributions. a. The binomial distribution b. The geometric distribution c. The Poisson distribution During World War I, the British government established the Industrial Fatigue Research Board (IFRB), later known as the Industrial Health Research Board (IHRB) (Haight 00). The board was created because of concern of the large number of accidental deaths and injuries in the British war roduction industries. One of the data sets the considered was the number of accidents exerienced b women working on 6- inch shells during the eriod Februar 3, 98, to March 0, 98. These are dislaed in the table below. Thus, 447 women had no accidents during this time eriod, but two had at least five accidents. Number of Accidents or more Frequenc Observed

99 99 a. Find the average number of accidents a woman had during this time eriod. (Assume all observations in the categor 5 or more are exactl 5.) b. After the work of von Bortkiewicz (see Section 4.8.3), the Poisson had been alied to a large number of random henomena and, with few excetions, had been found to describe the data well. This had led the Poisson distribution to be called the random distribution, a term that is still found in the literature. Thus, the mathematicians at the IFRB began b modeling these data using the Poisson distribution. Find the exected number of women having 0,,, 3, 4, and 5 accidents using the mean found in (a) and the Poisson distribution. How well do ou think this model describes the data? c. Greenwood and Woods (99) suggested fitting a negative binomial distribution to these data. Find the exected number of women having 0,,, 3, 4, and 5 accidents using the mean found in (a) and the geometric (negative binomial with r ) distribution. How well do ou think this model describes the data? Historical note: Researchers were uzzled as to wh the negative binomial fit better than the Poisson until, in 90, Greenwood and Yule suggested the following model. Suose that the robabilit an given woman will have an accident is distributed according to a Poisson distribution with mean λ. However, λ varies from woman to woman according to a gamma distribution (see Chater 5). Then the number of accidents would have a negative binomial distribution. The value λ of lambda associated with a woman was called her accident roneness The Deartment of Transortation s Federal Auto Insurance and Comensation Stud was based on a random samle of 7,84 California licensed drivers (Ferreira, 97). The number of accidents in which each was involved from November, 959, to Februar, 968, was determined. The summar results are given in the table below. Number of Frequenc Accidents Observed 0 5,47, or more 6 a. Find the average number of accidents a California licensed driver had during this time eriod. (Assume all observations in the categor 6 or more are exactl 6.) b. Find the exected number of drivers being involved in 0,,, 3, 4, 5 and 6 accidents using the mean found in (a) and the Poisson distribution. How well do ou think this model describes the data? c. Find the exected number of women having 0,,, 3, 4, 5 and 6 accidents using the mean found in (a) and the geometric (negative binomial with r ) distribution. How well do ou think this model describes the data?

100 00 Note: In both this and Exercise 4.37, a better fit of the negative binomial can be found b using a non-integer value for r. Although we continue to restrict our attention to onl integer values, the negative binomial distribution is well defined for an real value r > The sul office for a large construction firm has three welding units of Brand A in stock. If a welding unit is requested, the robabilit is 0.7 that the request will be for this articular brand. On a tical da, five requests for welding units come to the office. Find the robabilit that all three Brand A units will be in use on that da Refer to Exercise If the sul office also stocks three welding units that are not Brand A, find the robabilit that exactl one of these units will be left immediatel after the third Brand A unit is requested In the game Lotto 6-49, six numbers are randoml chosen, without relacement, from to 49. A laer who matches all six numbers, in an order, wins the jackot. a. What is the robabilit of winning an given jackot with one game ticket? b. If a game ticket costs $.00, what is the exected winnings from laing Lotto 6-49 once. c. Suose a erson bus one Lotto 6-49 ticket each week for a hundred ears. Assuming all ears have 5 weeks, what is the robabilit of winning at least one jackot during this time? (Hint: Use a Poisson aroximation) d. Given the setting in (c), what is the exected winnings over 00 ears? 4.4. The robabilit of a customer s arriving at a grocer service counter in an one second equals 0.. Assume that customers arrive in a random stream and, hence, that the arrival at an one second is indeendent of an other arrival. a. Find the robabilit that the first arrival will occur during the third -second interval. b. Find the robabilit that the first arrival will not occur until at least the third -second interval. c. Find the robabilit that no arrivals will occur in the first 5 seconds d. Find the robabilit that at least three eole will arrive in the first 5 seconds Of a oulation of consumers, 60% are reuted to refer a articular brand, A, of toothaste. If a grou of consumers are interviewed, find the robabilit of the following events. a. exactl five eole are interviewed before encountering a consumer who refers brand A b. at least five eole are interviewed before encountering a consumer who refers brand A The mean number of automobiles entering a mountain tunnel er -minute eriod is one. If an excessive number of cars enter the tunnel during a brief eriod of time, the result is a hazardous situation. a. Find the robabilit that the number of autos entering the tunnel during a -minute eriod exceeds three.

101 0 b. Assume that the tunnel is observed during ten -minute intervals, thus giving ten indeendent observations, Y, Y, Y 0, on a Poisson random variable. Find the robabilit that Y > 3 during at least one of the ten -minute intervals Suose that 0% of a brand of M3 laers will fail before their guarantee has exired. Suose 000 laers are sold this month, and let Y denote the number that will not fail during the guarantee eriod. a. Find the exected value and variance of Y. b. Within what limit would Y be exected to fail? (Hint: Use Tchebsheff s Theorerm.) a. Consider a binomial exeriment for n 0 and Calculate the binomial robabilities for Y 0,,, 3, and 4. b. Calculate the same robabilities, but this time use the Poisson aroximation with λ n. Comare the two results The manufacturer of a low-calorie dair drink wishes to comare the taste aeal of a new formula (B) with that of the standard formula (A). Each of four judges is given three glasses in random order; two containing formula A and the other containing formula B. Each judge is asked to choose which glass he most enjoed. Suose that the two formulas are equall attractive. Let Y be the number of judges stating a reference for the new formula. a. Find the robabilit function for Y. b. What is the robabilit that at least three of the four judges will state a reference for the new formula? c. Find the exected value of Y. d. Find the variance of Y Show that the hergeometric robabilit function aroaches the binomial in the limit as N and as r/n remains constant; that is, show that for constant r/n. r N r n r n lim N n r N q n A lot of N 00 industrial roducts contains 40 defectives. Let Y be the number of defectives in a random samle of size 0. Find (0) using the following distributions. a. The hergeometric robabilit distribution b. The binomial robabilit distribution Is N large enough so that the binomial robabilit function rovides a good aroximation to the hergeometric robabilit function? For simlicit, let us assume that there are two kinds of drivers. The safe drivers, who constitute 70% of the oulation, have a robabilit of 0. of causing an accident in

102 0 a ear. The rest of the oulation are accident makers, who have a robabilit of 0.5 of causing an accident in a ear. The insurance remium is $800 times one s robabilit of causing an accident in the following ear. A new subscriber has caused an accident during the first ear. What should her insurance remium be for the next ear? 4.5. A merchant stocks a certain erishable item. He knows that on an given da he will have a demand for two, three, or four of these items, with robabilities 0., 0.3, and 0.5, resectivel. He bus the items for $.00 each and sells them for $.0 each. An items left at the end of the da reresent a total loss. How man items should the merchant stock to maximize his exected dail rofit? 4.5. It is known that 5% of the oulation has disease A, which can be discovered b means of a blood test. Suose that N (a large number) eole are tested. This can be done in two was:. Each erson is tested searatel.. The blood samle of k eole are ooled together and analzed. (Assume that N nk, with n an integer.) If the test is negative, all of the ersons in the ool are health (that is, just this one test is needed). If the test is ositive, each of the k ersons must be tested searatel (that is, a total of k tests are needed.) a. For fixed k, what is the exected number of tests needed in method ()? b. Find the value for k that will minimize the exected number of tests in method (). c. How man tests does art (b) save in comarison with art (a)? Four ossible winning numbers for a lotter AC-673, FK-97, OJ-80, and PM-8 are given to ou. You will win a rize if one of our numbers marches one of the winning numbers. You are told that there is one first rize of $50,000; two second rizes of $75,000, and then third rizes of $000 each. All ou have to do is mail the couon back; no urchase is required. From the structure of the numbers ou have received, it is obvious that the entire list consists of all the ermutations of two letters from the alhabet, followed b four digits. Is the couon worth mailing back for 45 cents ostage? For a discrete random variable taking on values 0,,,, show that E( ) n 0 P( > n). References Bortkewitsch Das Gasetz der Kleinen Zahlen. Leizig. Boswell, M.T. and G.P. Patil Chance mechanisms generating the negative binomial distribution. In G.P. Patil (ed.). Random Counts in Scientific Work. Vol. Universit Park: Pennslvania State Universit. Chu, Singfat Using soccer goals to motivate the Poisson rocess. INFORMS Transactions on Education 3(): htt://ite.ubs.informs.org/vol3no/chu/.

103 03 Eves, Howard. (969). Introduction to the Histor of Mathematics, 3 rd Edition. New York: Holt, Rinehart and Winston. David, F. N. (955). Studies in the histor of robabilit and statistics. Biometrika 5: -5. Ferreira, Joseh Jr Designing Equitable Merit Rating Plans. Working Paer #OR Oerations Research Center, Massachusetts Institute of Technolog. Folks, J. L. (98). Ideas of Statistics. New York: John Wile & Sons. Goranson, U.G. and J. Hall. (980). Airworthiness of long life jet transort structures. Aeronautical Journal 84: Greenwood, M. and H.M. Woods. 99. The Incidence of Industrial Accidents uon Individuals with Secific Reference to Multile Accidents. Industrial Fatigue Research Board. Reort No. 4. London. Greenwood, M. and C.V. Yule. 90. An inquir into the nature of frequenc distributions reresentative of multile haenings, with articular reference to the occurrence of multile attacks of disease or reeated accidents. Journal of the Roal Statistical Societ 89: Haight, Frank A. 00. Accident Proneness: The Histor of an Idea. Technical Reort #UCIITS-WP-0-4. Institute of Transortation Studies, Universit of California, Irvine. Sakamoto, C.M Alication of the Poisson and Negative Binomial models to thunderstorms and hail das robabilities in Nevada. Monthl Weather Review 0: Schmuland, B. 00. Shark attacks and the Poisson aroximation, π in the Sk, Issue 4, htt:// Kemen, J.G. and J.L. Snell Finite Markov Chains. Sringer-Verlag. Kenned, William and James Gentle Statistical Comuting. New York: Marcel Dekker, Inc. Stigler, Stehen M The Histor of Statistics: The Measurement of Uncertaint before 900. Cambridge, Massachusetts: The Belkna Press of Harvard Universit Press. Student On the error of counting with a haemactometer. Biometrika 5: Whitaker, Luc. 95. On the Poisson law of large numbers. Biometrika 0: 36-7.

104 Yang, Mark C.K. and David H. Robinson Understanding and Learning Statistics b Comuter. World Scientific. 04