11. Let X be a bivariate random variable having the probability density N(m, S), with m ¼ 5 8

and S ¼ 2 1 1 3

:

(a) Define the regression curve of X1 on X2 . What is E(X1 | x2 ¼ 9)?

(b) What is the conditional variance of X1 given that x2 ¼ 9?

(c) What is the probability that x1 > 5? What is the probability that x1 > 5, given that x2 ¼ 9?

In problems 12–20 below, identify the most appropriate parametric family of density functions from those presented in this chapter on which to base the probability space for the experiment described, and answer the questions using the probability space you define: