= Random-sum transform identity
{title2=$M_S(t)=G_N(M_X(t))$}
For <independent and identically distributed random variables> $X_j$ independent of a nonnegative integer count $N$, set $S=\sum_{j=1}^N X_j$, including $S=0$ when $N=0$. Conditioning on the count gives $\mathbb E[e^{tS}\mid N=k]=M_X(t)^k$, hence the displayed composition of the <moment-generating function> with the <probability generating function>. For positive summands the identity holds for all $t\le0$; positive $t$ requires convergence of the composed series. This permits <Laplace transforms of nonnegative random variables> to identify the law even when positive <exponential moments> do not exist.
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