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Unformatted text preview: Notes on the Negative Binomial Distribution John D. Cook October 28, 2009 Abstract These notes give several properties of the negative binomial distri- bution. 1. Parameterizations 2. The connection between the negative binomial distribution and the binomial theorem 3. The mean and variance 4. The negative binomial as a Poisson with gamma mean 5. Relations to other distributions 6. Conjugate prior 1 Parameterizations There are a couple variations of the negative binomial distribution. The first version counts the number of the trial at which the r th success occurs. With this version, P ( X 1 = x | p,r ) = x- 1 r- 1 ! p r (1- p ) x- r . for integer x ≥ r . Here 0 < p < 1 and r is a positive integer. The second version counts the number of failures before the r th success. With this version, P ( X 2 = x | p,r ) = r + x- 1 x ! p r (1- p ) x . 1 for integer x ≥ 0. If X 1 is a negative binomial random variable according to the first definition, then X 2 = X 1- r is a negative binomial according to the second definition. We will standardize on this second version for the remainder of these notes. One advantage to this version is that the range of x is non-negative integers. Also, the definition can be more easily extended to all positive real values of r since there is no factor of r in the bottom of the binomial coefficient....
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This note was uploaded on 02/15/2012 for the course GEO 6938 taught by Professor Staff during the Summer '08 term at University of Florida.
- Summer '08