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Formula-Sheet-431S08

# Formula-Sheet-431S08 - Formula Sheet for ACTSC 431/831 Fall...

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Formula Sheet for ACTSC 431/831 – Fall 2008 1. Discrete Distributions (a) Poisson with parameter λ > 0: A random variable X is said to have a Poisson distribution denoted by X P ( λ ) if X has the following probability function (pf): Pr { X = k } = λ k e - λ k ! , k = 0 , 1 , 2 , ··· with E ( X ) = V ar ( X ) = λ ; the mgf M X ( t ) = E ( e tX ) = exp { λ ( e t - 1) } ; and the pgf P X ( z ) = E ( z X ) = exp { λ ( z - 1) } . (b) Binomial with parameters n (positive integer) and 0 < p < 1: A random variable X is said to have a binomial distribution denoted by X b ( n, p ) if X has the following pf: Pr { X = k } = ± n k ! p k (1 - p ) n - k , k = 0 , 1 , ··· ,n with E ( X ) = np ; V ar ( X ) = np (1 - p ); M X ( t ) = (1 + p ( e t - 1)) n ; and P X ( z ) = (1 + p ( z - 1)) n . (c) Bernoulli with parameter 0 < p < 1 is a special binomial distribution when n = 1, i.e. b (1 , p ). (d)

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Formula-Sheet-431S08 - Formula Sheet for ACTSC 431/831 Fall...

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