Let X, Y be independent exponential random variables with means 1 and 2 respectively.

Let

Z = 1, if X < Y

Z = 0, otherwise

Find E(X|Z) and V(X|Z).

We should first find E(X|Z=z)

E(X|Z=z) = integral (from 0 to inf) of xf(x|z).

However, how do we find f(x|z) ?