A point P is chosen uniformly in an n-dimensional sphere of radius 1. Next, a point Q is chosen uniformly within the concentric sphere, centered at the origin, going through P . Let X and Y be the distances of P and Q, respectively, to the common center. Find the joint density function of X and Y and the conditional expectations E(Y | X = x) and E(X | Y = y).
Hint 1. Begin by trying the case n = 2.
Hint 2. The volume of an n-dimensional sphere of radius r is equal to cnrn, where cn is some constant (which is of no interest for the problem). Remark. For n = 1 we rediscover the stick from Example 2.1.