Suppose that X has the Poisson distribution with mean t, and that Y has the gamma distribution with parameters = k and =

Suppose that X has the Poisson distribution with mean λt, and that Y has the gamma distribution with parameters α = k and β = λ, where k is a positive integer. Show that Pr(X ≥ k) = Pr(Y ≤ t) by showing that both the left side and the right side of this equation can be regarded as the probability of the same event in a Poisson process in which the expected number of occurrences per unit of time is λ.