Let U1, U2 be independent random variables both uniformly distributed on [0, 1], and set M = max(U1, U2) and N = min(U1, U2).
Find the conditional joint density of (U1, U2) given M ≤ 1/2
I know that the joint density of (U1, U2) is 1 since it's a square. But given the condition M, visually it's like restricting the U's to a square 1/4 the size. So conditional joint density of (U1, U2) when M = 1/2 should be 4. But I'm not sure how to account for M less than or equal to 4.