Problem source

A candidate finishes examination questions in time $T$, where $T$ has probability density function 
\[
\mathrm{f}(t)=t\mathrm{e}^{-t}\qquad t\geqslant0,
\]
the probabilities for the various questions being independent. Find the moment generating function of $T$ and hence find the moment generating function for the total time $U$ taken to finish two such questions.
Show that the probability density function for $U$ is 
\[
\mathrm{g}(u)=\frac{1}{6}u^{3}\mathrm{e}^{-u}\qquad u\geqslant0.
\]
Find the probability density function for the total time taken to answer $n$ such questions.

Solution source

\begin{align*}
&& M_T(x) &= \mathbb{E}[e^{xT}] \\
&&&= \int_0^{\infty} e^{xt}te^{-t} \d t \\
&&&= \int_0^{\infty}te^{(x-1)t} \d t \\
&&&= \left [ \frac{t}{x-1} e^{(x-1)t} \right]_0^{\infty} - \int_0^\infty \frac{e^{(x-1)t}}{x-1} \d t \\
&&&= \left [ \frac{e^{(x-1)t}}{(x-1)^2} \right]_0^{\infty} \\
&&&= \frac{1}{(x-1)^2} \\
\\
&& M_U(x) &= M_{T_1+T_2}(x) \\
&&&= \frac1{(x-1)^4} \\
\\
&& I_n &= \int_0^{\infty} t^ne^{(x-1)t} \d t \\
&&&= \left[ \frac{1}{(x-1)}t^ne^{(x-1)t} \right]_0^{\infty} - \frac{n}{(x-1)} \int_0^{\infty}t^{n-1}e^{(x-1)t} \d t \\
&&&= -\frac{n}{(x-1)}I_{n-1} \\
\Rightarrow && I_n &= \frac{n!}{(1-x)^{n+1}} \\
\\
\Rightarrow && \int_0^{\infty} e^{xt} \frac16u^3e^{-u} \d u &= \int_0^{\infty} \frac16u^3e^{(x-1)u} \d u \\
&&&=  \frac{1}{(1-x)^4} \\
\Rightarrow && f_U(u) &= \frac16u^3e^{-u} \\
\\
&& M_{X_1+\cdots+X_n}(x) &= \frac{1}{(x-1)^{2n}} \\
\Rightarrow && f_{X_1+\cdots+X_n}(t) &= \frac1{(2n-1)!} t^{2n-1}e^{-t}
\end{align*}

(NB: This is the gamma distribution)