1. Show that, if X and Y are random variables taking on only two values each, and if E(XY ) = E(X)E(Y ), then X and Y are independent.
2. A royal family has children until it has a boy or until it has three children, whichever comes first. Assume that each child is a boy with probability 1/2. Find the expected number of boys in this royal family and the expected num- ber of girls.