Chapter 11 Probability Models for Counts Copyright © 2011 Pearson Education, Inc 11.1 Random Variables for Counts How many doctors should management expect a pharmaceutical rep to meet in a day if only 40% of visits reach a doctor? Is a rep who meets or more doctors in a day doing exceptionally well? Need a discrete random variable to model counts and provide a method for finding probabilities of 30 Copyright © 2011 Pearson Education, Inc 11.1 Random Variables for Counts Bernoulli Random Variable Bernoulli trials are random events with three characteristics: Two possible outcomes (success, failure) Fixed probability of success (p) Independence of 30 Copyright © 2011 Pearson Education, Inc 11.1 Random Variables for Counts Bernoulli Random Variable - Definition A random variable B with two possible values, = success and = failure, as determined in a Bernoulli trial E(B) = p Var(B) = p(1-p) of 30 Copyright © 2011 Pearson Education, Inc 11.1 Random Variables for Counts Counting Successes (Binomial) Y, the sum of iid Bernoulli random variables, is a binomial random variable Y = number of success in n Bernoulli trials (each trial with probability of success = p) Defined by two parameters: n and p of 30 Copyright © 2011 Pearson Education, Inc 11.1 Random Variables for Counts Counting Successes (Binomial) We can define the number of doctors seen by a pharmaceutical rep in 10 visits as a binomial random variable This random variable, Y, is defined by n = 10 visits and p = 0.40 (40% success in reaching a doctor) of 30 Copyright © 2011 Pearson Education, Inc 11.2 Binomial Model Assumptions Using a binomial random variable to describe a real phenomenon 10% Condition: if trials are selected at random, it is OK to ignore dependence caused by sampling from a finite population if the selected trials make up less than 10% of the population of 30 Copyright © 2011 Pearson Education, Inc 11.3 Properties of Binomial Random Variables Mean and Variance E(Y) = np Var(Y) = np(1 - p) of 30 Copyright © 2011 Pearson Education, Inc 11.3 Properties of Binomial Random Variables Pharmaceutical Rep Example E(Y) = np = (10)(0.40) = We expect a rep to see doctors in 10 visits Var(Y) = np(1 - p) = (1)(0.40)(0.60) = 2.4 SD(Y) = 1.55 A rep who has seen doctors has performed 2.6 standard deviations above the mean 10 of 30 Copyright © 2011 Pearson Education, Inc 4M Example 11.1: FOCUS ON SALES Motivation A focus group with nine randomly chosen participants was shown a prototype of a new product and asked if they would buy it at a price of $99.95 Six of them said yes The development team claimed that 80% of customers would buy the new product at that price If the claim is correct, what results would we expect from the focus group? 16 of 30 Copyright © 2011 Pearson Education, Inc 4M Example 11.1: FOCUS ON SALES Method Use the binomial model for this situation Each focus group member has two possible responses: yes, no We can use X ~ Bi(n = 9, p = 0.8) to represent the number of yes responses out of nine 17 of 30 Copyright © 2011 Pearson Education, Inc 4M Example 11.1: FOCUS ON SALES Mechanics – Find E(X) and SD(X) E(X) = np = (9)(0.8) = 7.2 Var(X) = np(1-p) = (9)(0.8)(0.2) = 1.44 SD(X) = 1.2 The expected number is higher than the observed number of 18 of 30 Copyright © 2011 Pearson Education, Inc 4M Example 11.1: FOCUS ON SALES Mechanics – Probability Distribution While is not the most likely outcome, it is still common 19 of 30 Copyright © 2011 Pearson Education, Inc 4M Example 11.1: FOCUS ON SALES Message The results of the focus group are in line with what we would expect to see if the development team’s claim is correct 20 of 30 Copyright © 2011 Pearson Education, Inc 11.4 Poisson Model A Poisson Random Variable Describes the number of events determined by a random process during an interval of time or space Is not finite (possible values are infinite) Is defined by λ (lambda), the rate of events 21 of 30 Copyright © 2011 Pearson Education, Inc 11.4 Poisson Model The Poisson Probability Distribution P( X = x ) = e −λ λ x! x x = 0, 1, 2, E(X) = λ Var(X) = λ 22 of 30 Copyright © 2011 Pearson Education, Inc 11.4 Poisson Model The Poisson Model Uses a Poisson random variable to describe counts of data Is appropriate for situations like • The number of calls arriving at the help desk in a 10-minute interval • The number of imperfections per square meter of glass panel 23 of 30 Copyright © 2011 Pearson Education, Inc 4M Example 11.2: DEFECTS IN SEMICONDUCTORS Motivation A supplier claims that its wafers have defect per 400 cm2 Each wafer is 20 cm in diameter, so the area is 314 cm2 What is the mean number of defects and the standard deviation? 24 of 30 Copyright © 2011 Pearson Education, Inc 4M Example 11.2: DEFECTS IN SEMICONDUCTORS Method The random variable is the number of defects on a randomly selected wafer The Poisson model applies 25 of 30 Copyright © 2011 Pearson Education, Inc 4M Example 11.2: DEFECTS IN SEMICONDUCTORS Mechanics – Find λ The assumed defect rate is per 400 cm Since a wafer has an area of 314 cm 2, λ = 314/400 = 0.785 E(X) = 0.785 SD(X) = 0.886 P(X = 0) = 0.456 26 of 30 Copyright © 2011 Pearson Education, Inc 4M Example 11.2: DEFECTS IN SEMICONDUCTORS Message The chip maker can expect about 0.8 defects per wafer About 46% of the wafers will be defect free 27 of 30 Copyright © 2011 Pearson Education, Inc Best Practices Ensure that you have Bernoulli trials if you are going to use the binomial model Use the binomial model to simplify the analysis of counts Use the Poisson model when the count accumulates during an interval 28 of 30 Copyright © 2011 Pearson Education, Inc Best Practices (Continued) Check the assumptions of a model Use a Poisson model to simplify counts of rare events 29 of 30 Copyright © 2011 Pearson Education, Inc Pitfalls Do not presume independence without checking Do not assume stable conditions routinely 30 of 30 Copyright © 2011 Pearson Education, Inc ... a discrete random variable to model counts and provide a method for finding probabilities of 30 Copyright © 2 011 Pearson Education, Inc 11. 1 Random Variables for Counts Bernoulli Random Variable.. .Chapter 11 Probability Models for Counts Copyright © 2 011 Pearson Education, Inc 11. 1 Random Variables for Counts How many doctors should management... 30 Copyright © 2 011 Pearson Education, Inc 11. 1 Random Variables for Counts Counting Successes (Binomial) Y, the sum of iid Bernoulli random variables, is a binomial random variable Y