Hint 1: It may be useful to know that, for two random variables U and V with finite

variance, the variance of the sum is given by

V ar(U + V ) = V ar(U) + V ar(V ) + 2E[UV ] 2E[U]E[V ].

Hint 2: For random variables X1, . . . ,Xn with finite expectations, we have E[X1 + . . . +

Xn] = E[X1] + . . . + E[Xn].)

M coins are randomly distributed into N piggy banks. Denote by Y the number of piggy

banks with at least one coin.

a) Find expected value E[Y]. If you cannot do it for general M,N.

b) Set Xi = 1 if piggy bank i has at least one coin, and Xi = 0 if it has no

coins. Find E[XiXj ], including the case i = j.

c) Assume now N = 2. Compute variance Var[Y ].