Chapter 5 Discrete Probability Distributions

Transcription

1 Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide Chapter 5 Discrete Probability Distributios Radom Variables Discrete Probability Distributios Epected Value ad Variace Poisso Distributio Hypergeometric Distributio Slide 2 Radom Variables A radom variable is a umerical descriptio of the outcome of a eperimet. A discrete radom variable may assume either a fiite umber of values or a ifiite sequece of values. A cotiuous radom variable may assume ay umerical value i a iterval or collectio of itervals. Slide 3

2 Eample: JSL Appliaces Discrete radom variable with a fiite umber of values Let = umber of TVs sold at the store i oe day, where ca take o 5 values (0,, 2, 3, 4 Slide 4 Eample: JSL Appliaces Discrete radom variable with a ifiite sequece of values Let = umber of customers arrivig i oe day, where ca take o the values 0,, 2,... We ca cout the customers arrivig, but there is o fiite upper limit o the umber that might arrive. Slide 5 Radom Variables Questio Radom Variable Type Family size Distace from home to store Ow dog or cat = Number of depedets reported o ta retur = Distace i miles from home to the store site = if ow o pet; = 2 if ow dog(s oly; = 3 if ow cat(s oly; = 4 if ow dog(s ad cat(s Discrete Cotiuous Discrete Slide 6 2

3 Discrete Probability Distributios The probability distributio for a radom variable describes how probabilities are distributed over the values of the radom variable. We ca describe a discrete probability distributio with a table, graph, or equatio. Slide 7 Discrete Probability Distributios The probability distributio is defied by a probability fuctio, deoted by f(, which provides the probability for each value of the radom variable. The required coditios for a discrete probability fuctio are: f( > 0 Σf( = Slide 8 Discrete Probability Distributios Usig past data o TV sales, a tabular represetatio of the probability distributio for TV sales was developed. Number Uits Sold of Days f( /200 Slide 9 3

4 Discrete Probability Distributios Graphical Represetatio of Probability Distributio Probability Values of Radom Variable (TV sales Slide 0 Discrete Uiform Probability Distributio The discrete uiform probability distributio is the simplest eample of a discrete probability distributio give by a formula. The discrete uiform probability fuctio is f( = / the values of the radom variable are equally likely where: = the umber of values the radom variable may assume Slide Epected Value ad Variace The epected value, or mea, of a radom variable is a measure of its cetral locatio. E( = µ = Σf( The variace summarizes the variability i the values of a radom variable. Var( = σ 2 = Σ( - µ 2 f( The stadard deviatio, σ, is defied as the positive square root of the variace. Slide 2 4

5 Epected Value ad Variace Epected Value f( f( E( =.20 epected umber of TVs sold i a day Slide 3 Epected Value ad Variace Variace ad Stadard Deviatio µ ( - µ 2 f( ( - µ 2 f( Variace of daily sales = σ 2 =.660 Stadard deviatio of daily sales =.2884 TVs TVs squared Slide 4 Four Properties of a Biomial Eperimet. The eperimet cosists of a sequece of idetical trials. 2. Two outcomes, success ad failure, are possible o each trial. 3. The probability of a success, deoted by p, does ot chage from trial to trial. statioarity 4. The trials are idepedet. assumptio Slide 5 5

6 Our iterest is i the umber of successes occurrig i the trials. We let deote the umber of successes occurrig i the trials. Slide 6 Biomial Probability Fuctio! f ( = p ( p!(! ( where: f( = the probability of successes i trials = the umber of trials p = the probability of success o ay oe trial Slide 7 Biomial Probability Fuctio! f ( = p ( p!(! (!!(! Number of eperimetal outcomes providig eactly successes i trials p ( p ( Probability of a particular sequece of trial outcomes with successes i trials Slide 8 6

7 Eample: Evas Electroics Evas is cocered about a low retetio rate for employees. I recet years, maagemet has see a turover of 0% of the hourly employees aually. Thus, for ay hourly employee chose at radom, maagemet estimates a probability of 0. that the perso will ot be with the compay et year. Slide 9 Usig the Biomial Probability Fuctio Choosig 3 hourly employees at radom, what is the probability that of them will leave the compay this year? Let: p =.0, = 3, =! f ( p ( p ( =!(! 3! f ( = (0. (0.9 = 3(.(.8 =.243!(3! 2 ( (0. (0.9 3(.( Slide 20 Tree Diagram st Worker 2 d Worker 3 rd Worker Prob. L ( Leaves (. S ( Leaves (. L ( Stays (.9 S ( Stays (.9 Leaves (. Stays (.9 L (. S (.9 L (. S ( Slide 2 7

8 Usig Tables of Biomial Probabilities p Slide 22 Epected Value Variace E( = µ = p Var( = σ 2 = p( p Stadard Deviatio σ = p ( p Slide 23 Epected Value E( = µ = 3(. =.3 employees out of 3 Variace Var( = σ 2 = 3(.(.9 =.27 Stadard Deviatio σ = 3(.(.9 =.52 employees Slide 24 8

9 Poisso Distributio A Poisso distributed radom variable is ofte useful i estimatig the umber of occurreces over a specified iterval of time or space It is a discrete radom variable that may assume a ifiite sequece of values ( = 0,, 2,.... Slide 25 Poisso Distributio Eamples of a Poisso distributed radom variable: the umber of kotholes i 4 liear feet of pie board the umber of vehicles arrivig at a toll booth i oe hour Slide 26 Poisso Distributio Two Properties of a Poisso Eperimet. The probability of a occurrece is the same for ay two itervals of equal legth. 2. The occurrece or ooccurrece i ay iterval is idepedet of the occurrece or ooccurrece i ay other iterval. Slide 27 9

10 Poisso Distributio Poisso Probability Fuctio µ µ e f ( =! where: f( = probability of occurreces i a iterval µ = mea umber of occurreces i a iterval e = Slide 28 Poisso Distributio Eample: Mercy Hospital Patiets arrive at the MERCY emergecy room of Mercy Hospital at the average rate of 6 per hour o weeked eveigs. What is the probability of 4 arrivals i 30 miutes o a weeked eveig? Slide 29 Poisso Distributio MERCY Usig the Poisso Probability Fuctio µ = 6/hour = 3/half-hour, = ( f (4 = =.680 4! Slide 30 0

11 Poisso Distributio MERCY Usig Poisso Probability Tables µ Slide 3 Poisso Distributio MERCY Poisso Distributio of Arrivals 0.25 Poisso Probabilities Probability Number of Arrivals i 30 Miutes actually, the sequece cotiues:, 2, Slide 32 Poisso Distributio A property of the Poisso distributio is that the mea ad variace are equal. µ = σ 2 Slide 33

12 Poisso Distributio MERCY Variace for Number of Arrivals Durig 30-Miute Periods µ = σ 2 = 3 Slide 34 Hypergeometric Distributio The hypergeometric distributio is closely related to the biomial distributio. However, for the hypergeometric distributio: the trials are ot idepedet, ad the probability of success chages from trial to trial. Slide 35 Hypergeometric Distributio Hypergeometric Probability Fuctio where: r N r f ( = for 0 < < r N f( = probability of successes i trials = umber of trials N = umber of elemets i the populatio r = umber of elemets i the populatio labeled success Slide 36 2

13 ZAP Hypergeometric Distributio Hypergeometric Probability Fuctio r N r f ( = for 0 < < r N umber of ways failures ca be selected umber of ways from a total of N r failures successes ca be selected i the populatio from a total of r successes i the populatio umber of ways a sample of size ca be selected from a populatio of size N Slide 37 Hypergeometric Distributio Eample: Neveready Bob Neveready has removed two dead batteries from a flashlight ad iadvertetly migled them with the two good batteries he iteded as replacemets. The four batteries look idetical. Bob ow radomly selects two of the four batteries. What is the probability he selects the two good batteries? ZAP ZAP ZAP Slide 38 Hypergeometric Distributio Usig the Hypergeometric Fuctio r N r 2 2 2! 2! 2 0 2!0! 0!2! f( = = = = =.67 N 4 4! 6 2 2!2! where: = 2 = umber of good batteries selected = 2 = umber of batteries selected N = 4 = umber of batteries i total r = 2 = umber of good batteries i total Slide 39 3

14 Hypergeometric Distributio Mea r E ( = µ = N Variace 2 r r N Var ( = σ = N N N Slide 40 Hypergeometric Distributio Mea r 2 µ = = 2 = N 4 Variace σ = 2 = = Slide 4 Hypergeometric Distributio Cosider a hypergeometric distributio with trials ad let p = (r/ deote the probability of a success o the first trial. If the populatio size is large, the term (N /(N approaches. The epected value ad variace ca be writte E( = p ad Var( = p( p. Note that these are the epressios for the epected value ad variace of a biomial distributio. cotiued Slide 42 4

15 Hypergeometric Distributio Whe the populatio size is large, a hypergeometric distributio ca be approimated by a biomial distributio with trials ad a probability of success p = (r/n. Slide 43 Ed of Chapter 5 Slide 44 5