Table A: Important Discrete Distributions distribution pmf mean variance mgf Poisson e−λ λx x! λ λ e−λ+λet Binomial ` n x ´ πx(1 − π)n−x nπ nπ(1 − π) (πet + 1 − π)n Geometric(1) π(1 − π)x 1 π − 1 1 − π π2 π 1 − (1 − π)et Negative Binomial(1) ` s+x−1 x ´ πs(1 − π)x s π − s s(1 − π) π2 » π 1 − (1 − π)et –s Hypergeometric(2) ` m x ´` n k−x ´ ` m+n k ´ Nπ Nπ(1 − π) N −k N −1 Notes: (1) Some texts define the geometric and negative binomial distributions differ- ently. Here X counts the number of failures before s successes (s = 1 for the geometric random variables). Some authors prefer a random variable XT that counts the total number of trials. This is simply a shifting of the distribution by a constant (the number of successes): XT = X + s. The formulas for the pdf, mean, variance, and moment generating function of XT all follow easily from this equation. (2) For the hypergeometric distribution, the parameters are m items of the type being counted, n items of the other type, and k items selected without re- placement. We define N = m + n (total number of items) and π = m N = m m+n (proportion of items that are of the first type).
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