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?