The phrase “A stick is broken at random into three pieces” can be interpreted in several ways. Let us identify the stick with interval [0, 1] and let 0

are defined as in (i). (iii) X is chosen from [0, 1] according to the uniform distribution, and then Y is chosen with uniform distribution on [X, 1]. (iv) U is chosen with uniform distribution on [0, 1]. Next, one of the intervals [0, U] or [U, 1] is chosen at random, with probability U and 1 − U,respectively. Then V is chosen with uniform distribution from the chosen interval, and again, X = min(U, V ) and Y = max(U, V ).

In each of the cases, (i)–(iv) find the joint density of (X, Y ) and the marginal densities of X and Y . Which of the ways (i)–(iv) are equivalent?