# Probability of an exponential random variable being greater than a gamma random variable?

Let V have exponential(a) density, and let W be independent of V with gamma(s,b) density. Find P(V>W).

What I did for this problem is I integrated the conditional probability P(V>W|W)f(w)dw from w = 0 to infinity, where f(w) represents the density function of W. After integrating I arrived at the answer b^s / (a+b)^s.

I received 2 out of 6 points on my final exam, and the notes say I need to do a double integral over the triangular region where V > W. However, aren't these equivalent? When I compute the double integral I arrive at the same answer. Is it invalid to integrate the conditional density here?

• It seems to me that you are correct; the double integral is implicit in your use of $P(V>W|W)$. It may be that when you wrote the answer, you weren't totally clear about using the CDF (actually, 1 - the CDF) instead of the density itself, or perhaps the grader wanted you to be explicit about the double integral, for reasons which escape me personally. – jbowman May 16 '17 at 19:59
• It wouldn't be the first time a grader has missed that an answer that's different from a supplied solution may still be correct. – Glen_b -Reinstate Monica May 17 '17 at 0:23