Moment-generating function of a gamma distribution ($M_X(t)={\left(\lambda /(\lambda -t)\right)}^r,t<\lambda$)

For the rate-parameterized Gamma distribution with shape $r>0$ and rate $\lambda >0$, integration of its probability density giveswhen $t<\lambda$, by the gamma function integral and a change of variables. The integral is infinite for $t\ge \lambda$. 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.