Chapter 6, Problem 6.21TE

Suppose that W, the amount of moisture in the air on a given day, is a gamma random variable with parameters $\left(t,\beta \right)$. That is, its density is $f\left(\omega \right)=\frac{\beta e^{-\beta \omega }{\left(\beta \omega \right)}^{t-1}}{\Gamma \left(t\right)},\omega >0$. Suppose also that given that $W=\omega$. the number of accidents during that day—call it N—has a Poisson distribution with mean $\omega$. Show that the conditional distribution of

W given that $N=n$ is the gamma distribution with parameters $\left(t+n,\beta +1\right)$.