1. A fair coin is tossed four times. Let X be the number of heads obtained on the rst three tosses of the coin. Let Y be the number of heads on all four tosses of the coin. a) Find the joint probability distribution of X and Y. ( (b) Find the mean and variance of X. (c) Find the conditional probability distribution of Y given that X = 2. ((1) Find the mean of the conditional probability distribution of Y given that X = 2. 3. The random variable X has a discrete uniform distribution with values 1, 2 and 3, i.e. P(X = 2') = 1/3 for i = 1, 2, 3. The random variable Y has a discrete uniform distribution With values 1, 2, 3 and 4, i.e. P(Y : i) : 1/4 for 2' : 1, 2, 3, 4. X and Y are independent. (a) Derive the probability distribution of X + Y. (b) What are E(X + Y) and Var(X + Y)? 5.* Suppose that X and Y are random variables, and a, b, c and d are constants. (a) Show that: Cov(aX + b, CY + d) : ac: Cov(X, Y). (b) Derive Corr(aX + b, CY + d). ((3) Suppose that Z = cX+d, where c and d are constants. Using the result you obtained in (b), or in some other way, show that: Corr(X,Z) = 1 for c > O and: Corr(X, Z) = 71 for c