Chapter 5 Discrete Probability Distributions 1 Chapter Outline

Chapter 5 Discrete Probability Distributions 1

Chapter Outline Ø Ø Random Variables Discrete Probability Distributions Expected Value and Variance Binomial Probability Distribution 2

Random Variables § A random variable is a numerical description of the outcome of an experiment. § A discrete random variable assumes numerical values that have gaps or jumps between them. § A continuous random variable assumes numerical values that have NO gaps or jumps between them. 3

Random Variables Experiment Take a quiz with 10 Ture/False questions Random Variable x x = Number of correct Type Discrete answers Run 5 K x = time to finish a 5 k run Continuous Weigh a sample of 36 cans of coffee (labeled as 3 lbs x = the average weight of a Continuous sample of 36 cans of coffee 4

Discrete Probability Distributions § The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. § We can describe a discrete probability distribution with a table, graph, or formula. 5

Discrete Probability Distributions § The probability distribution is defined by a probability function, denoted by f(x), which provides the probability for each value of the random variable. § The required conditions for a discrete probability function are: f (x ) > 0 f (x ) = 1 6

Discrete Probability Distributions § Example: Probabilities of the # of correct answers to a quiz of 4 True/False questions. v We can use a table to represent the probability distribution. v The random variable x represents the number of correct answers. x 0 1 2 3 4 f (x ). 10. 25. 35. 20. 10 1. 00 7

Discrete Uniform Probability Distribution The discrete uniform probability distribution is the simplest example of a discrete probability distribution given by a formula. The discrete uniform probability function is f(x) = 1/n where: the values of the random variable are equally likely n = the number of values the random variable may assume 8

Expected Value § The expected value, or mean, of a random variable is a measure of its central location. E(x) = = xf(x) § The expected value is a weighted average of the values the random variable may assume. The weights are the probabilities. § The arithmetic mean introduced in Chapter 3 can be viewed as a special case of weighted average, where the weights for all the values are the same, i. e. 1/n. 9

Variance and Standard Deviation 1. The variance summarizes the variability in the values of a random variable. Var(x) = 2 = (x - )2 f(x) 1. The variance is a weighted average of the squared deviations of a random variable from its mean. The weights are the probabilities. 2. The standard deviation, , is defined as the positive square root of the variance. 10

Expected Value § Example: Take a quiz (# of correct answers) x 0 1 2 3 4 f (x ) xf(x). 10. 00. 25. 35. 70. 20. 60. 10. 40 E(x) = 1. 95 Expected number of correct answers 11

Variance § Example: Take a quiz (# of correct answers) x x- (x - )2 f (x ) (x - )2 f(x) 0 1 2 3 4 -1. 95 -0. 95 0. 05 1. 05 2. 05 3. 8025 0. 9025 0. 0025 1. 1025 4. 2025 . 10. 25. 35. 20. 10 . 3803. 2256. 0009. 2205. 4203 Variance of daily sales = 2 = 1. 2476 Standard deviation of daily sales = = 1. 11 correct answers 12

Binomial Probability Distribution Properties of a Binominal Experiment: 1. The experiment consists of a sequence of n identical trials; 2. Only two outcomes, success and failure, are possible on each trial; 3. The probability of a success, denoted by p, does not change from trial to trial; 4. The trials are independent from one another. 13

Binomial Probability Distribution § Our interest is in the number of successes occurring in the n trials. § We let x denote the number of successes occurring in the n trials. § Either outcome can be named as ‘Success’. We need to make sure that in the calculation, the probability p is matched with the definition of the random variable x. 14

Binomial Probability Distribution § Binomial Probability Function where: x = the number of successes p = the probability of a success on one trial n = the number of trials f(x) = the probability of x successes in n trials n! = n(n – 1)(n – 2) …. . (2)(1) 15

Binomial Probability Distribution § Binomial Probability Function Number of experimental outcomes providing exactly x successes in n trials Probability of a particular sequence of trial outcomes with x successes in n trials 16

Binomial Probability Distribution § Example: Purchasing a pair of shoes Based on recent sales data, a shoe store manager estimates that the probability a customer makes a purchase is 30%. For the next three customers, what is the probability that exactly 1 of them will make a purchase? Analysis: Is this example a binomial experiment? If so, which outcome is to be named ‘Success’? And what is the probability of Success? 17

Binomial Probability Distribution § Example: Purchasing a pair of shoes Does the example satisfy the properties of a binomial distribution? n N trials? – Yes, 3 trials ( 3 customers) § Two outcomes for each trial? – Yes, purchase or not § Probability of success stays the same – 30% chance for making a purchase can be assumed to be the same for all the customers. § Independent trials – Assume three customers are independent in their decision on making a purchase. 18

Binomial Probability Distribution § Example: Purchasing a pair of shoes How many favorable outcomes are there where exactly ONE of the next three customers makes a purchase? With the success representing ‘making a purchase’ and the three customers assumed to be independent, we should have the following outcomes and their probabilities: Probability of Experimental Outcome (S, F, F) (F, S, F) (F, F, S) p(1 – p) = (. 3)(. 7) =. 147 (1 – p)p(1 – p) = (. 7)(. 3)(. 7) =. 147 (1 – p)p = (. 7)(. 3) =. 147 Total =. 441 19

Binomial Probability Distribution § Example: Purchasing a pair of shoes Let: p =. 30, n = 3, x = 1 Using the probability function 20

Binomial Probability Distribution Example: Purchasing a pair of shoes 1 st Customer 2 nd Customer P (. 3) Purchase (. 3) Using a tree diagram 3 rd Customer P (. 3) x 3 Prob. . 027 NP (. 7) P (. 3) 2 . 063 1 . 147 P (. 3) 2 . 063 NP (. 7) P (. 3) 1 . 147 0 . 343 NP (. 7) NP(. 7) P (. 3) Not Purchase (. 7) NP (. 7) 21

Binomial Probability Table § Statisticians have developed tables that give probabilities and cumulative probabilities for a binomial random variable. In the appendix of our textbook, you can locate the binomial probability tables. § For our example, the table is presented as below (where x represents the number of success): x 0 1 2 3 f (x ). 343. 441. 189. 027 1. 00 22

Binomial Probability Distribution § We can apply the formulas of expected value and variance for a binomial probability distribution. However, those formulas can be further simplified as follows: § Expected Value E(x) = = np § Variance Var(x) = 2 = np(1 - p) § Standard Deviation 23

Binomial Probability Distribution § Example: Purchasing a pair of shoes § Expected Value E(x) = np = 3(. 3) =. 9 customers out of 3 § Variance Var(x) = np(1 – p) = 3(. 3)(. 7) =. 63 § Standard Deviation customers 24