5 42 chapter 2 discrete type random variables so the

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Unformatted text preview: n−r . Therefore, the pmf of Sr is given by p(n) = n−1 r p (1 − p)n−r r−1 for n ≥ r. This is called the negative binomial distribution with parameters r and p. To check that the pmf sums to one, begin with the MacLaurin series expansion (i.e. Taylor series expansion about zero) of (1 − x)−r : ∞ (1 − x)−r = k=0 k+r−1 k x. r−1 Set x = 1 − p and use the change of variables k = n − r to get: ∞ n=r n−1 r p (1 − p)n−r = pr r−1 ∞ k=0 k+r−1 (1 − p)k = 1. r−1 Use of the expansion of (1 − x)−r here, in analogy to the expansion of (1 + x)n used for the binomial distribution, explains the name “negative binomial distribution.” Since Sr = L1 + . . . Lr , where 1 r each Lj has mean p , E [Sr ] = p . It is shown in Example 4.8.1 that Var(Sr ) = rVar(L1 ) = r(1−p) . p2 2.7. THE POISSON DISTRIBUTION–A LIMIT OF BERNOULLI DISTRIBUTIONS 2.7 41 The Poisson distribution–a limit of Bernoulli distributions By definition, the Poisson probability distribution with parameter λ > 0 is the one w...
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