Solution
= Solution
The <moment-generating function> of a real <random variable> $X$ is
$$
\boxed{M_X(t)=\mathbb E[e^{tX}]}
$$
for real $t$ where the expectation is finite. One may regard it as an extended nonnegative function elsewhere. In particular $M_X(0)=1$; finiteness on a neighborhood of zero is the usual hypothesis for recovering moments and using its local expansion.