5.5 Exercises

EXERCISE 5.1   Consider $X\sim N_{2}(\mu,\Sigma)$ with $\mu=(2,2)^{\top}$ and $\Sigma ={\displaystyle \left( {1 \atop 0}\ {0 \atop 1} \right)}$ and the matrices $\data{A} = {\displaystyle {1 \choose 1}^{\top}}$, $\data{B}=
{\displaystyle {1 \choose -1}^{\top}}$. Show that $\data{A}X$ and $\data{B}X$ are independent.

EXERCISE 5.2   Prove Theorem 5.4.

EXERCISE 5.3   Prove proposition (c) of Theorem 5.7.

EXERCISE 5.4   Let

\begin{displaymath}X\sim N_{2}\left( \left( \begin{array}{c} 1\\ 2 \end{array} \...
...ft( \begin{array}{cc}
2 & 1\\ 1 & 2 \end{array} \right)\right)\end{displaymath}

and

\begin{displaymath}Y\mid X\sim N_{2} \left( \left( \begin{array}{c}X_1\\ X_1+X_2...
...t( \begin{array}{cc} 1 & 0\\ 0 & 1 \end{array} \right) \right).\end{displaymath}

a)
Determine the distribution of $Y_2\mid Y_1$.
b)
Determine the distribution of $W=X-Y$.

EXERCISE 5.5   Consider $\left( \begin{array}{c}X\\ Y\\ Z \end{array} \right) \sim N_3(\mu,\Sigma).$ Compute $\mu$ and $\Sigma$ knowing that

\begin{eqnarray*}
Y\mid Z & \sim & N_1(-Z,1)\\
\mu_{Z\mid Y} & = & -\frac{1}{3}-\frac{1}{3}Y\\
X\mid Y,Z & \sim & N_1(2+2Y+3Z,1)
.\end{eqnarray*}



Determine the distributions of $X \mid Y$ and of $X \mid Y+Z$.

EXERCISE 5.6   Knowing that

\begin{eqnarray*}
Z & \sim & N_1(0,1)\\
Y\mid Z & \sim & N_1(1+Z,1)\\
X\mid Y,Z & \sim & N_1(1-Y,1)\\
\end{eqnarray*}



a)
find the distribution of $\left( \begin{array}{c} X\\ Y\\ Z \end{array} \right)$ and of $Y\mid {X,Z}$.
b)
find the distribution of

\begin{displaymath}\left( \begin{array}{c}
U\\ V \end{array} \right) =
\left( \begin{array}{c}
1+Z\\ 1-Y \end{array} \right).\end{displaymath}

c)
compute $E(Y \mid {U=2})$.

EXERCISE 5.7   Suppose $\left( \begin{array}{c} X \\ Y \end{array} \right) \sim N_2 (\mu, \Sigma)$ with $\Sigma$ positive definite. Is it possible that
a)
$\mu_{X \mid Y}=3Y^2$,
b)
$\sigma_{XX \mid Y}=2+Y^2$,
c)
$\mu_{X \mid Y}=3-Y$, and
d)
$\sigma_{XX \mid Y}=5$ ?

EXERCISE 5.8   Let $X \sim N_3 \left( \left( \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right), \lef...
...} 11 & {-6} & 2 \\ {-6} & 10 & {-4} \\ 2 & {-4} & 6 \end{array} \right) \right)$.
a)
Find the best linear approximation of $X_3$ by a linear function of $X_1$ and $X_2$ and compute the multiple correlation between $X_3$ and $(X_1,X_2)$.
b)
Let $Z_1=X_2-X_3,\ Z_2=X_2+X_3$ and $(Z_3 \mid {Z_1,Z_2}) \sim N_1(Z_1+Z_2,10)$. Compute the distribution of $\left( \begin{array}{c} Z_1 \\ Z_2 \\ Z_3 \end{array} \right)$.

EXERCISE 5.9   Let $(X, Y, Z)^\top$ be a trivariate normal r.v. with

\begin{eqnarray*}Y \mid Z & \sim & N_1(2Z,24)\\
Z \mid X & \sim & N_1(2X+3,14)\\
X & \sim & N_1(1,4)\\
\textrm{and}\ \rho_{XY} & = & 0.5.
\end{eqnarray*}



Find the distribution of $(X, Y, Z)^\top$ and compute the partial correlation between $X$ and $Y$ for fixed $Z$. Do you think it is reasonable to approximate $X$ by a linear function of $Y$ and $Z$?

EXERCISE 5.10   Let $X \sim N_4 \left( \left( \begin{array}{c} 1\\ 2\\ 3\\ 4 \end{array} \right),
\...
...\ 1 & 4 & 2 & 1\\ 2 & 2 & 16 & 1\\
4 & 1 & 1 & 9 \end{array} \right) \right).$
a)
Give the best linear approximation of $X_2$ as a function of $(X_1,X_4)$ and evaluate the quality of the approximation.
b)
Give the best linear approximation of $X_2$ as a function of $(X_1,X_3,X_4)$ and compare your answer with part a).

EXERCISE 5.11   Prove Theorem 5.2.
(Hint: complete the linear transformation $Z= \left( \data{A} \atop {\data{I}}_{p-q}\right)X +
\left( c \atop 0_{p-q}\right)$ and then use Theorem 5.1 to get the marginal of the first $q$ components of $Z$.)

EXERCISE 5.12   Prove Corollaries 5.1 and 5.2.