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Q18Section I (Objective (Q1–Q80))18 of 80

Probability — moment generating function of a random sum

Let X1,X2,…X_1,X_2,\dots be a sequence of i.i.d. random variables and SN=X1+X2+⋯+XNS_N=X_1+X_2+\dots+X_N, where NN is a random variable independent of XiX_i, i=1,2,3,…i=1,2,3,\dots. If MX1(t)=E(etX1)M_{X_1}(t)=E(e^{tX_1}), then the moment generating function of SNS_N is

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