Let U, V be two uniform independent random variables on [0, 2]. (a) Given U = 1, find the conditional expectation of 3U + 4V. (b) Given U = 1, find the conditional expectation of 3eU+V. (c) Given U = 1, find the conditional variance of 3U +4V.
(d) Given U + V = 3, find the conditional expectation of U - V and 3U +4V. (Hint: consider the map g(u, v) = (u - v, u + v).)