Solution
= Solution
This is the <Erlang distribution> with shape $n$ and rate one. Direct integration gives, for $t<1$,
$$
\boxed{M_X(t)=\frac1{(n-1)!}\int_0^\infty x^{n-1}e^{-(1-t)x}dx
=(1-t)^{-n}.}
$$
= Solution
This is the <Erlang distribution> with shape $n$ and rate one. Direct integration gives, for $t<1$,
$$
\boxed{M_X(t)=\frac1{(n-1)!}\int_0^\infty x^{n-1}e^{-(1-t)x}dx
=(1-t)^{-n}.}
$$