 Random-sum transform identity

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 $t\le 0$; 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.