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Chapter 5 Discrete Random Variables McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter Outline 5.1 5.2 5.3 5.4 Two Types of Random Variables Discrete Probability Distributions The Binomial Distribution The Poisson Distribution (Optional) 5-2 5.1 Two Types of Random Variables Random variable: a variable that assumes numerical values that are determined by the outcome of an experiment Discrete Continuous Discrete random variable: Possible values can be counted or listed The number of defective units in a batch of 20 A rating on a scale of 1 to 5 5-3 Random Variables Continued Continuous random variable: May assume any numerical value in one or more intervals The waiting time for a credit card authorization The interest rate charged on a business loan 5-4 5.2 Discrete Probability Distributions The probability distribution of a discrete random variable is a table, graph or formula that gives the probability associated with each possible value that the variable can assume Notation: Denote the values of the random variable by x and the value’s associated probability by p(x) 5-5 Discrete Probability Distribution Properties px 0 for each valueof x p x 1 allx 5-6 Expected Value of a Discrete Random Variable The mean or expected value of a discrete random variable X is: m X x px All x m is the value expected to occur in the long run and on average 5-7 Variance The variance is the average of the squared deviations of the different values of the random variable from the expected value The variance of a discrete random variable is: 2 X x m X px 2 All x 5-8 Standard Deviation The standard deviation is the positive square root of the variance The variance and standard deviation measure the spread of the values of the random variable from their expected value 5-9 Example Table 5.2 5-10 Example Continued m x All x xpx 0.03 1.20 2.50 3.20 4.05 5.02 2.1 All x x m x px 2 2 x 0 2.1 0.03 1 2.1 0.20 2 2.1 0.50 2 2 2 3 2.1 0.20 4 2.1 0.05 5 2.1 0.02 2 .89 2 2 5-11 5.3 The Binomial Distribution The binomial experiment… Experiment consists of n identical trials 2. Each trial results in either “success” or “failure” 3. Probability of success, p, is constant from trial to trial 4. Trials are independent 1. If x is the total number of successes in n trials of a binomial experiment, then x is a binomial random variable 5-12 Binomial Distribution Continued n! x n- x px = p q x!n - x ! 5-13 Example 5.9 5-14 Binomial Probability Table p = 0.1 values of p (.05 to .50) x 0 1 2 3 4 0.05 0.8145 0.1715 0.0135 0.0005 0.0000 0.95 0.1 0.6561 0.2916 0.0486 0.0036 0.0001 0.9 0.15 0.5220 0.3685 0.0975 0.0115 0.0005 0.85 … … … … … … … 0.50 0.0625 0.2500 0.3750 0.2500 0.0625 0.50 4 3 2 1 0 x values of p (.05 to .50) P(x = 2) = 0.0486 Table 5.7 (a) 5-15 Several Binomial Distributions Figure 5.6 5-16 Mean and Variance of a Binomial Random Variable m x np npq 2 x X npq 2 x 5-17 Example m x np 8.95 7.6 npq 8.95.05 .38 2 x X .38 .6164 2 x 5-18 5.4 The Poisson Distribution (Optional) Consider the number of times an event occurs over an interval of time or space, and assume that The probability of occurrence is the same for any intervals of equal length 2. The occurrence in any interval is independent of an occurrence in any non-overlapping interval 1. If x = the number of occurrences in a specified interval, then x is a Poisson random variable 5-19 The Poisson Distribution px e m m x! Continued x 5-20 Poisson Probability Table Table 5.9 5-21 Poisson Probability Calculations Table 5.10 5-22 Mean and Variance of a Poisson Random Variable Mean mx = m Variance 2x = m Standard deviation x is square root of variance 2x 5-23 Several Poisson Distributions Figure 5.9 5-24