Partial answers are fine

Problem 2 (15p). Let a > 0 be fixed, and let X, Y be independent random variables such that X has the geometric distribution with p = 1 -e and Y has the exponential distribution with parameter a. For c ER, compute the probability P(X + Y > c) in terms of c and [c] = infine ZIn > c). Hint: The above means that P(X 2 0) = P(Y 2 0) = 1, X only takes values in {0, 1, ...}, and for n E (0, 1, ...}, c' E [0, 0o), P(X = n) = (1 - e"")e"an, P(X > n) = e-an, P(Y > c') = e-ac Solution. 0