Chapter 6, Problem 6.22TE

Let W be a gamma random variable with parameters $\left(t,\beta \right)$ and suppose that conditional on $W=\omega ,X_1,X_2,\mathrm{...}{X}_n$ are independent exponential random variables with rate $\omega$. Show that the conditional distribution of W given that $X_1=x_1,X_2=x_2,\mathrm{...},X_n=x_n$ is gamma with parameters $\left(t+n,\beta +{\displaystyle \underset{i=1}{\overset{n}{\sum }}{x}_i}\right)$.