Solomon (1983) details the following biological model. Suppose that each of a random number, N, of insects lays Xi eggs, where the Xts are independent, identically distributed random variables. The total number of eggs laid is H = X1 +. . . . . . .+ XN. What is the distribution of H? It is common to assume that N is Poisson(λ). Furthermore, if we assume that each Xi has the logarithmic series distribution (see Exercise 3.14) with success probability p, we have the hierarchical model

Show that the marginal distribution of H is negative binomial(r, p), where r = -λ/log(p). (It is easiest to calculate and identify the mgf of H using Theorems 4.4.3 and 4.6.7. Stuart and Ord 1987, Section 5.21, also mention this derivation of the logarithmic series distribution. They refer to Zf as a randomly stopped sum.)

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-1 1-p log(P) N~Poisson(A)