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Chapter 5: Discrete Probability Distributions

GBS221, Class 18429 March 19, 2012 Notes Compiled by Nicolas C. Rouse, Instructor, Phoenix College

Chapter Objectives

1. Understand the concepts of a random variable and a probability distribution. 2. Be able to distinguish between discrete and continuous random variables. 3. Be able to compute and interpret the expected value, variance, and standard deviation for a discrete random variable and understand how an Excel worksheet can be used to ease the burden of the calculations. 4. Be able to compute probabilities using a binomial probability distribution and be able to compute these probabilities using Excel's BINOMDIST function. 5. Be able to compute probabilities using a Poisson probability distribution and be able to compute these probabilities using Excel's POISSON function. 6. Be able to compute probabilities using a hypergeometric probability distribution and be able to compute these probabilities using Excel's HYPGEOMDIST function.

1. and 2. Random Variable and Probability Distribution/Discrete and Continuous Random Variables

· · · A random variable is a numerical description of the outcome of an experiment. A random variable can be classified as being either discrete or continuous depending on the numerical values it assumes. A discrete random variable may assume either a finite number of values or an infinite sequence of values. For example, if we contact 5 customers, the random variable's possible values are 0,1,2,3,4,5 if we are looking for customers who place an order. If we are looking for gender of a specific customer, the possible values can be 0 if male or 1 if female (or any other numeric value we assign each gender). A continuous random variable may assume any numerical value in an interval or collection of intervals. Experimental values based on measurement scales such as time, weight, distance, and temperature can be described by continuous random variables. An example of this could be the time in customer arrivals in minutes where x >= 0. We could look at the number of ounces that a soft drink can is filled where 0 <= x <= 12.1. Discrete Probability Distributions o The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. o The probability distribution is defined by a probability function, denoted by f(x), which provides the probability for each value of the random variable. . o The required conditions for a discrete probability function are: · f(x) >= 0 · f(x) = 1 o We can describe a discrete probability distribution with a table, graph, or equation. Discrete Uniform Probability Distribution o The discrete uniform probability distribution is the simplest example of a discrete probability distribution given by a formula. o The discrete uniform probability function is · f(x) = 1/n where: n = the number of values the random variable may assume o Note that the values of the random variable are equally likely.

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3. Expected Value, Variance, and Standard Deviation for Discrete Random Variables

· Expected Value and Variance o The expected value, or mean, of a random variable is a measure of its central location. o Expected value of a discrete random variable: · E(x) = µ = xf(x) o The variance summarizes the variability in the values of a random variable. o Variance of a discrete random variable: 2 2 · Var(x) = = (x - µ) f(x)

Chapter 5: Discrete Probability Distributions

GBS221, Class 18429 March 19, 2012 Notes Compiled by Nicolas C. Rouse, Instructor, Phoenix College o o The standard deviation, , is defined as the positive square root of the variance. The standard deviation is measured in the same units as the random variable and therefore is often preferred in describing variability.

4. Binomial Probability Distribution (FOR EXCEL FUNCTIONS, SEE PAGES 222-223)

· Binomial Probability Distribution o The binomial probability distribution is associated with a multiple-step experiment that we call the binomial experiment. o A binomial experiment has the following four properties: · The experiment consists of a sequence of n identical trials. · Two outcomes, success and failure, are possible on each trial. · The probability of a success, denoted by p, does not change from trial to trial (this property is called the stationarity assumption). · The trials are independent. o A special mathematical formula, called the binomial probability function, can be used to compute the probability of x successes in n trials. Binomial Probability Function o The function can be viewed as consisting of two parts. · Part 1: Number of experiments; outcomes providing exactly x successes in n trials is: · n!/x!(n - x)! · Part 2: Probability of a particular sequence of trial outcomes with x successes in n trials is: · pX (1- p)(n-x) · Putting the two parts together, we get the binomial probability function:

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Expected Value and Variance o Expected Value: E(x) = µ = np 2 o Variance: Var(x) = = np(1 ­ p)

5. Poisson Probability Distribution

· · The Poisson probability distribution is often useful in estimating the number of occurrences over a specified interval of time or space. A Poisson experiment has the following two properties: o The probability of an occurrence is the same for any two intervals of equal length. o The occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval.

Chapter 5: Discrete Probability Distributions

GBS221, Class 18429 March 19, 2012 Notes Compiled by Nicolas C. Rouse, Instructor, Phoenix College · The Poisson probability function is:

6. Hypergeometric Probability Distribution

· · · · The hypergeometric distribution is closely related to the binomial distribution. With the hypergeometric distribution, the trials are not independent, and the probability of success changes from trial to trial. The hypergeometric probability function is used to compute the probability that in a random sample of n elements, selected without replacement, we will obtain x elements labeled success and n - x elements labeled failure. The hypergeometric probability function is:

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Expected Value and Variance for Hypergeometric distribution:

Chapter 5: Discrete Probability Distributions

GBS221, Class 18429 March 19, 2012 Notes Compiled by Nicolas C. Rouse, Instructor, Phoenix College · The function can be viewed as consisting of three parts. o Part 1: Number of ways a sample of size n can be selected from a population of size N.

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Part 2: Number of ways that x successes can be selected from a total of r successes in the population.

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Part 3: Number of ways that n - x failures can be selected from a total of N - r failures in the population.

Chapter 5: Discrete Probability Distributions

GBS221, Class 18429 March 19, 2012 Notes Compiled by Nicolas C. Rouse, Instructor, Phoenix College

KEY TERMS

A numerical description of the outcome of an experiment. A random variable that may assume either a finite number of values or an infinite sequence of values. Continuous random variable A random variable that may assume any numerical value in an interval or collection of intervals. Probability distribution A description of how the probabilities are distributed over the values of the random variable. Probability function A function, denoted by f (x), that provides the probability that x assumes a particular value for a discrete random variable. Discrete uniform probability distribution A probability distribution for which each possible value of the random variable has the same probability. Expected value A measure of the central location of a random variable. Variance A measure of the variability, or dispersion, of a random variable. Standard deviation The positive square root of the variance. Binomial experiment An experiment having the four properties stated at the beginning of Section 5.4. Binomial probability distribution A probability distribution showing the probability of x successes in n trials of a binomial experiment. Binomial probability function The function used to compute binomial probabilities. Poisson probability distribution A probability distribution showing the probability of x occurrences of an event over a specified interval of time or space. Poisson probability function The function used to compute Poisson probabilities. Hypergeometric probability distribution A probability distribution showing the probability of x successes in n trials from a population with r successes and N - r failures. Hypergeometric probability function The function used to compute hypergeometric probabilities. Random variable Discrete random variable

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