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Unformatted text preview: Chapter Three Discrete Random Variables and Probability Distributions 3.1 Random Variables A random variable is any rule which associates a number with each outcome in the sample space S . Alternatively, it may be defined as a variable whose value is a numerical outcome of a random phe nomenon. The term variable indicates that different values are possible and the term random indicates that probabilities are associated with the events. Random variables are usually denoted by capital letters of the alphabet, such as X or Y . A simple random variable is the Bernoulli random variable or binary random variable whose possible values are 0 and 1. A random variable X may take on a finite or infinite number of possible values. A random variable X is said to be discrete if its set of possible values is finite or if its elements can be listed as a sequence. Examples of discrete random variables are: X = if the result of a fair coin toss is tails 1 if the result of a fair coin toss is heads and Y = the number of people in this room born in the month of September, with possible values , 1 , 2 ,...,n , where n is the number of people in the room. Example (Devore, Page 93, Exercise 7) For each of the following random variables, describe the set of possible values for the variable and state whether the variable is discrete: a. X = the number of unbroken eggs in a randomly chosen standard egg carton. Solution X is a discrete random variable with possible values 0 , 1 , 2 ,..., 12. b. Y = the number of time a duffer has to swing at a golf ball before hitting it. Solution Y is a discrete random variable with possible values 1 , 2 , 3 ,... . c. Z = the tension (psi) at which a randomly selected tennis racket has been strung. Solution Since tension can be measured to any decimal place, there is an infinite number of possible values between the minimum and maximum tensions and Z is not discrete (it is continuous) with possible values { z : min Z z max Z } . 3.2 Probability Distributions for Discrete Random Variables The assignment of probabilities to the values of a random variable X is called the probability distribution of X . For a discrete random variable, the probability distribution is often called the probability mass function or pmf . For every possible realized value x of the random variable X , the pmf specifies the probability of observing that value when the experiment is performed. Hence p ( x ) = P ( X = x ) is the probability that the random variable X takes on the value x . The pmf is a mass function because each of the possible values of the discrete random variable X has the mass of probability P ( X = x ) associated with it. The pmf is alternatively called the discrete probability density function (pdf). A function p ( x ) is a valid pmf if: p ( x ) 0 for all x , and 1 X all x p ( x ) = 1....
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This note was uploaded on 01/08/2012 for the course EXST 4050 taught by Professor Staff during the Fall '10 term at LSU.
 Fall '10
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