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Unformatted text preview: 1 Topic 3  Discrete distributions Basics of discrete distributions  pages 81  84 Mean and variance of a discrete distribution  pages 93  95 , 97 Binomial distribution  pages 8589 , 95  96 , 98 Poisson distribution and process  pages 104 , 106  108 2 A random variable is a function which maps each element in the sample space of a random process to a numerical value. A discrete random variable takes on a finite or countable number of values. We will identify the distribution of a discrete random variable X by its probability mass function (pmf) , f X (x) = P ( X = x ). Requirements of a pmf: f ( x ) 0 for all possible x all ( ) 1 x f x = 3 Cumulative Distribution Function The cumulative distribution function (cdf) is given by An increasing function starting from a value of 0 and ending at a value of 1. When we specify a pmf or cdf, we are in essence choosing a probability model for our random variable. all ( ) ( ) ( ) t x F x P X x f t = = 4 Reliability example Consider the series system with three independent components each with reliability p . Let X i be 1 if the i th component works (S) and 0 if it fails (F). X i is called a Bernoulli random variable . Let f Xi (x) = P ( X i = x ) be the pmf for X i . f Xi (1) = f Xi (0) = p p p 5 Reliability example continued What is the pmf for X ? 3 1 Let be th e n u m ber of com ps. th at work i i X X = = Outcome X 1 X 2 X 3 X Probability SSS SSF SFS FSS FFF FFS FSF SFF x f X (x) 6 Reliability example continued Plot the pmf for X for p = 0.5. Plot the cdf for p = 0.5. 7 Reliability example continued What is the probability there are at most 2 working components if p = 0.5?...
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This note was uploaded on 03/19/2012 for the course MATH MATH 304 taught by Professor Young during the Spring '09 term at Texas A&M.
 Spring '09
 young

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