= Transform composition for an independent random sum
{title2=$M_Z(t)=G_N(M_Y(t))$}
For independent identically distributed summands, independent also of the count $N$, conditioning on $N=n$ gives $\mathbb E[e^{tZ}\mid N=n]=M_Y(t)^n$. The <law of total expectation> gives the composition of the count's <probability generating function> with the summand's <moment-generating function>. This identity is valid as an extended nonnegative expectation; finiteness near zero requires additional exponential-integrability assumptions.
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