4. (30pts) Charlie and Debbie decide to have children until either they have their rst daughter or they have I: children, Where k 2 1 is a xed integer. Assume that each child is either a boy or girl independently with probability 1/2 and that there are no multiple births. (a) Let X denote the number of sons they have. Find the PMF of X. (b) Let Y denote the number of daughters they have. Find the PMF of Y. (c) What is the expected number of daughters that they have? What is the expected number of sons that they have? ((1) Now suppose Charlie and Debbie simply decide to keep having children until they have their rst daughter. What is the distribution and PMF of X? (Hint: You may use the fact that 21:111' - 2_(i+1) = 1 2"\"(k + 1)