Let X and Y be independent and uniformly distributed on (0,1). Let M = min(X, Y) and N = max(X, Y).
(a) Find the joint density of M and N. (For 0 < m < n < 1, show that M ≤ m and N > n if and only if either {X ≤ m and Y > n} or {Y ≤ m and X >n}.)
(b) Find the conditional density of N given M = m and describe the conditional distribution.