Section Random Variables

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1 MAT 0 - Introduction to Statitic Section 5. - Random Variable A random variable i a variable that take on different numerical value which are determined by chance. Example 5. pg. 33 For each random experiment, define a random variable and identify the poible value of the variable. a. It i aumed that the larget number of children i 6, count the number of children within a family. b. Select a radial tire from the production line and determine the life of the tire, auming no tire ha ever lated le than 0,000 mile or more than 5,000 mile. c. Count the number of head in two toed of a fair coin. A dicrete random variable i a random variable that can take on a finite or countable number of value. A continuou random variable i continuou if the value of the random variable can aume any value or an uncountable number of value between any two poible value of the variable. Example 5. pg. 34 For each random variable, tate whether it i dicrete or continuou. a. The life of a 3 inch color picture tube. b. The number of phone call arriving at a college witchboard during the day. c. The number of employee that are abent from work at a teel manufacturing plant during a ummer day. d. The thickne of a multivitamin. 5.3 Probability Ditribution of a Dicrete Random Variable A probability ditribution i a ditribution which diplay the probabilitie aociated with all the poible value of a random variable. Characteritic of a Probability Ditribution of a Dicrete Random Variable. The probability aociated with a particular value of a dicrete random variable of a probability ditribution i alway a number between 0 and incluive.. The um of all the probabilitie of a probability ditribution mut alway be equal to one. Example 5.3 on pg. 37 in the Text - Conider the experiment of toing a fair coin until a head appear or until the coin ha been toed three time, whichever come firt. a. Contruct a probability ditribution for the number of tail. b. Contruct a probability hitogram for thi probability ditribution. c. Uing the probability hitogram, what i the probability of getting no tail?

2 MAT 0 - Introduction to Statitic Here i the experiment: to a fair coin until a head appear or until the coin ha been toed 3 time. Firt lit all poible outcome - there are four poible outcome: H T H T T H T T T Head on the t to Experiment over! Tail on the t to and Head on the nd to Experiment over! Tail on the t two toe and a Head on the 3 rd to Experiment over! or Tail on all three toe Experiment over! Now what do we want to conider? The number of tail which are: 0 tail, tail, tail, or 3 tail Head on the t to (H) = 0 tail Tail on the t to and Head on the nd to (TH) = tail Tail on the t two toe and a Head on the 3 rd to (TTH) OR = tail Tail on all three toe (TTT) = 3 tail Probability of getting a head i ½ and of getting a tail i ½ Letting X = the number of tail, we can contruct the following table: Number of Tail X Poible Outcome 0 H T H T T H 3 T T T Probability P(X) 4 Notice that the probabilitie add up to : 4 Now let contruct the probability ditribution for part a.

3 MAT 0 - Introduction to Statitic a. Probability Ditribution of Number of Tail Number of Tail X 0 3 Probability P(X) 4 b. The probability hitogram i contructed by caling the value of X along the horizontal axi and P(X) along the vertical axi. Probability Hitogram of Number of Tail c. From the hitogram, the probability of getting no tail i equal to the height of the bar graph pertaining to 0 tail which i ½ or 0.5. So the probability i ½ or 0.5. Review Example 5.4 on pg. 39 in the Text 3

4 MAT 0 - Introduction to Statitic To calculate the mean and tandard deviation for a probability ditribution we will ue the graphing calculator a follow: Enter all value of the random variable, X in Lit Enter the probabilitie, P(X) in Lit Ue -Var Stat Like thi: -Var Stat L, L Example 5.3 on pg. 37 in the Text PARTS ADDED by me! - Conider the experiment of toing a fair coin until a head appear or until the coin ha been toed three time, whichever come firt. NOT in text: You will be given the Probability Ditribution Table! THIS i how the quetion will begin! Probability Ditribution of Number of Tail Number of Tail X 0 3 Probability P(X) 4 d. What i the mot likely number of tail? X value with the larget probability, or 0 tail. e. What i the average number of tail toed for thi experiment? Mean, or 0.75 f. What i the tandard deviation? Standard Deviation i.05 Review Example 5.7 on pg in the Text Ue the calculator to find the mean and tandard deviation. 4

5 MAT 0 - Introduction to Statitic 5.5 Binomial Probability Ditribution A binomial experiment atifie the following four condition:. There are n identical trial. A binomial ditribution i the reult of a probability experiment that ha been repeated a predetermined number of n time, and each repetition (trial) of the experiment i identical - Such a toing a coin twenty time. The n identical trial are independent. Each outcome (trial) i independent and mutually excluive - For example: In the experiment of toing a coin n time, the outcome of each to (trial) i independent of any other to 3. The outcome for each trial can be claified a either a ucce or a failure. Each outcome i claified in one of two poibilitie, ucce or fail - The determination a to whether an outcome i a ucce or failure i a function on how the quetion i aked - Succe generally mean a poitive repone to the quetion See top of pg the to of a coin i either a head or a tail - the election of a poible anwer for a quetion on a multiple choice tet i either correct or incorrect - the to of a die reult in an outcome which i either a 5 or not a 5 - a new drug will either be effective or not effective 4. The probability of a ucce i the ame for each trial. The probability of ucce i the ame for each trial - Meaning, in a coin toing experiment, the probability of landing on Head i the ame for each to (trial) of the coin Binomial Probability Formula For a binomial experiment, the probability of getting uccee in n trial i computed uing the binomial probability formula. Thi formula i written a: ( n ) P ( uccee in n trial) = nc p q where: n= number of independent trial = number of uccee (n ) = number of failure nc = the number of way uccee can occur in n trial p = the probability of a ucce for one trial q = the probability of a failure for one trial = p Or you can ue the built-in function of your TI3/4 calculator:,, binompdf n p nd DISTR 0: binompdf ( n, p, ) ENTER 5

6 MAT 0 - Introduction to Statitic Example 5.9 on pg. 45 Conider the experiment of toing a fair coin five time, where we are intereted in getting a head. Can thi experiment be claified a a binomial experiment? (Doe it atify the four condition?) Review Example 5.0 and 5. on pg Example 5.4 pg Ue the binomial probability formula ( n ) P ( uccee in n trial) = C p q and the calculator, to determine the probability of getting 3 uccee in 4 trial if p = 3 n Example (NOT in text) Anwer the following if we conider number from 0 to 0? a. More than 7 b. Le than 3 c. At mot 4 d. At leat 6 Example (NOT in text) A fair coin i toed 0 time, where a ucce i getting a tail on a ingle to of the coin. Calculate the probability of: a. Getting three tail ( = 3) b. Getting at mot tail a. n= number of independent trial = 0 = number of uccee = number of tail = 3 (n ) = number of failure = 0 3 = 7 nc = the number of way uccee can occur in n trial = 0 C 3 = 0 p = the probability of a ucce for one trial = ½ q = the probability of a failure for one trial = p = ½ = ½ P ( uccee in n trial) = n C p q ( n ) P(getting 3 tail) = C = = 0.7 OR = binompdf 0,, 3 = 0.7 6

7 MAT 0 - Introduction to Statitic b. Getting at mot tail - mean getting tail or le or 0 tail and tail P( at mot tail) = P(0 tail) + P( tail) = 0 = = C = = C OR = binompdf 0,, 0 + 0,, binompdf = 0.00 Example 5.5 pg Each year the FBI report the probability of a car being tolen. In a recent report, the FBI tate that the probability a new car will be tolen during the year i out of 75. If you and your three friend own new car, what i the probability that none of thee car will be tolen thi year? n= number of independent trial = total number of car = = number of uccee = number of tolen car = (n ) = number of failure = nc = the number of way uccee can occur in n trial = p = the probability of a ucce for one trial = probability of a car being tolen = q = the probability of a failure for one trial = p = ( n ) P ( uccee in n trial) = C p q n Example 5.6 pg.57 - A tudent i going to gue at the anwer to all quetion on a five quetion multiple choice tet where there are four choice for each quetion. Calculate the probability of: a. Gueing three correct anwer b. Gueing five correct anwer c. Gueing at mot two correct anwer d. Gueing at leat four correct anwer n= number of independent trial = number of quetion on the tet = = number of uccee = number of correct anwer = (n ) = number of failure = nc = the number of way uccee can occur in n trial = p = the probability of a ucce for one trial = probability of gueing a correct anwer = q = the probability of a failure for one trial = p = ( n ) P ( uccee in n trial) = C p q n 7

8 MAT 0 - Introduction to Statitic a. Gueing three correct anwer b. Gueing five correct anwer c. Gueing at mot two correct anwer d. Gueing at leat four correct anwer Review Example 5.7 pg. 60-6

Stats on the TI 83 and TI 84 Calculator

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