Question: Assume that the random variable X_A is normally distributed with mean and variance equal to 100 and 64 respectively. Assume also that the independent random variable X_B is also normally distributed with mean and variance equal to 120 and 4 respectively.

(a) Let Y=3X_B-X_A. Find E(Y) and Var(Y). Then compute the probability that Y will be greater than 250. Draw the probability density function of Y, clearly and neatly, and show the area that corresponds to this probability.

(b) Let M_A be the mean of a random sample of size 8 drawn from population A. Similarly let M_B be the mean of a random sample of size 4 drawn from population B. Finally, let W=M_B-M_A. Find E(W) and Var(W). Then compute the probability that W will be smaller than 17.