Suppose that W, the amount of moisture in the air on a given day, is a gamma random variable with parameters $\displaystyle (t, \beta)$

Suppose also that given that $\displaystyle W = w $, the number of accidents during that day - call it N - has a poisson distribution with mean $\displaystyle w $.

Show that the conditional distribution of W given that N = n is the gamma distribution with parameters $\displaystyle (t+n, \beta +\sum_{i = 1}^n x_i)$

I would like some help to write the formula for the second supposition, as it goes from a continuous rv to a discrete one

Thanks