5. For a randomly selected family with two parents and one male child and one female child, let X1 be the average height of the parents in inches. Let X2 be the height of the (adult) male child and X3 be the height of the (adult) female child. Assume that (X 1, X2, X 3) follow a multivariate normal distribution with mean and covariance matrix 65 102 %1()2 1102 ,u = 70 and E = $102 102 :102 60 5102 102 102 o (a) What are the marginal distributions of X2 and X3? What are their corre sponding means? 0 (b) What is the correlation between the average parent height and the height of the male child? what is the correlation between the average parent hight and the height of the female child? 0 (c) What is the correlation between the heights of the two children? 0 (d) Conditioned on the midparent height X1 = 75 (tall parents), what is the joint distribution of (X2, X3)? 0 (e) From your answer to (d), what is the mean height of the male child given X1 = 75? What is the mean height of the female child given X1 = 75? Compare them with your answer in (d) and provide possible reasons. 0 (f) From your answer to (d), what is the correlation between the heights of the children once we know X1 = 75? [This is called conditional correlation.] Comparing it with your answer in (c), are you surprised? Provide some possible explanations