Let X be a geometric variable with parameter p (0, 1). Given X = k, we have that Y is a Gamma variable with parameters k and > 0 (i.e., given X = k, Y is the sum of k independent exponential variables with rate ).

Compute P(X = k | Y = y) for y > 0, k 1.

Please simplify your answer as much as possible (i.e., do not leave integrals or summations in your answer).