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Unformatted text preview: MA/STAT416: Probability Lecture Notes Spring 2011 1 Chapter 6: Standard Discrete Distributions March 23, 2011 6 Poisson and Negative Binomial Distributions 6.4 The Negative Binomial Distribution Definition. Let X be the number of the first toss at which the r th success is obtained. Then X has a negative binomial distribution; X has NB ( r,p ), with the pmf P ( X = x ) = x- 1 r- 1 p r (1- p ) x- r , x = r,r + 1 , . Theorem. If X has NB ( r,p ), r 1. Then, E ( X ) = r p , and V ar ( X ) = r (1- p ) p 2 . Example 1 . A couple want to have children till they have two girls. Find the distribution, expected value and variance of the number of children they will have. Note. This is the negative binomial distribution with parameters r = 2 ,p = 1 2 . Example 2 . Exploratory oil drilling in a certain region will strike oil with a 20% proba- bility. Find the probability that the third strike of oil comes at the sixth drill; that the third strike comes on or before the sixth drill....
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