= Solution
The <moment-generating function> of a real <random variable> $X$ is
$$
\boxed{M_X(t)=\mathbb E[e^{tX}]}
$$
for real $t$ for which this <expected value> is finite. Because the integrand is nonnegative, the extended <expected value> is always defined, but it may equal $+\infty$. We always have $M_X(0)=1$; the usual uniqueness and convergence theorems require finiteness on an open <interval> containing zero. When this condition holds, derivatives at zero recover the <moments>, $M_X^{(r)}(0)=\mathbb E[X^r]$.
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