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Probability Generating Function

Let X1, X2,... be i.i.d. random variables with p.g.f Gx(s) and N be a r.v independent of X with p.g.f. Gn(s). Define S=X1+X2+...+XN. Find Var(S) in terms of expectations and variances of X and N.

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The solution contains the determination of variance of a random sum in terms of the expectations and variances of X and N.

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