# Question: Suppose that W the amount of moisture in the air

Suppose that W, the amount of moisture in the air on a given day, is a gamma random variable with parameters (t, Î²). That is, its density is f(w) = Î²eâˆ’Î²w(Î²w)tâˆ’1 / Î“(t), w > 0. Suppose also that given that W = w, the number of accidents during that dayâ€”call it Nâ€”has a Poisson distribution with mean w. Show that the conditional distribution of W given that N = n is the gamma distribution with parameters (t + n, Î² + 1).

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