= Moment-generating function of a gamma distribution
{title2=$M_X(t)=\left(\lambda/(\lambda-t)\right)^r,\quad t<\lambda$}
For the rate-parameterized <Gamma distribution> with shape $r>0$ and rate $\lambda>0$, integration of its <probability density> gives
$$
\mathbb E[e^{tX}]=\frac{\lambda^r}{\Gamma(r)}\int_0^\infty x^{r-1}e^{-(\lambda-t)x}\,dx=\left(\frac{\lambda}{\lambda-t}\right)^r
$$
when $t<\lambda$, by the <gamma function> integral and a <change of variables>. The integral is infinite for $t\geq\lambda$. In particular the <moment-generating function> is finite on an open interval containing zero, so the <uniqueness theorem for moment-generating functions> applies. Multiplication of these <moment-generating functions> proves <additivity of independent gamma distributions with a common rate>.
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