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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 nonnegative 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
 Staff

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