= Moment-generating function of an independent product
If $U$ and $X$ are independent real <random variable>[random variables], then the <moment-generating function> of their product is $M_{UX}(t)=\mathbb E_U[M_X(tU)]$ wherever the expression is finite. This follows by conditioning on $U$: the conditional law of $X$ is unchanged by <independence>, so $\mathbb E(e^{tUX}\mid U)=M_X(tU)$. For $U$ uniform on $(0,1)$, the formula becomes $M_{UX}(t)=\int_0^1M_X(tu)du$. If $M_X$ is finite on a neighborhood of zero, this product also has a finite <moment-generating function> on a neighborhood of zero, because the bounded factor $U$ keeps $tu$ between zero and $t$.
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