Let \(\boldsymbol{X}\) be a bivariate random variable having the probability density \(N(\boldsymbol{\mu}, \boldsymbol{\Sigma})\), with

\(\boldsymbol{\mu}=\left[\begin{array}{l}5 \\ 8\end{array}ight]\) and \(\boldsymbol{\Sigma}=\left[\begin{array}{rr}2 & -1 \\ -1 & 3\end{array}ight]\).

(a) Define the regression curve of \(X_{1}\) on \(X_{2}\). What is \(\mathrm{E}\left(X_{1} \mid x_{2}=9ight)\) ?

(b) What is the conditional variance of \(X_{1}\) given that \(x_{2}=9\) ?

(c) What is the probability that \(x_{1}>5\) ? What is the probability that \(x_{1}>5\), given that \(x_{2}=9\) ?

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: