2. Let X m Exp(/\\) for A > 0. Let Y : [Xi be the result of rounding up X to the nearest integer, so that e.g. [3.1] = [3.8] = 4. (a) Note that Y takes values in N, i.e. P(Y E N) : 1. Compute py, the probability mass function of Y. Hence show that Y follows a geometric distribution with parameter p : l (FA. [4] (b) Compute GX|y(a,lk) = P(X g a|Y = k) for any is E N and any a > 0. [4] (c) Differentiate GXJy((L|k) with respect to a and hence compute the expected value EleY ~_ 19] of X conditional on Y : k where k: E N. [3]