4.7 Exercises

EXERCISE 4.1   Assume that the random vector $Y$ has the following normal distribution: $ Y \sim N_p(0,\data{I})$. Transform it according to (4.49) to create $X\sim N(\mu,\Sigma)$ with mean $\mu=(3,2)^{\top}$ and $\Sigma=\left({1\atop -1.5}{-1.5\atop 4}\right)$. How would you implement the resulting formula on a computer?

EXERCISE 4.2   Prove Theorem 4.7 using Theorem 4.5.

EXERCISE 4.3   Suppose that $X$ has mean zero and covariance $\Sigma=\left({1\atop 0}
{0\atop 2}\right)$. Let $Y=X_{1}+X_{2}$. Write $Y$ as a linear transformation, i.e., find the transformation matrix $\data{A}$. Then compute $Var(Y)$ via (4.26). Can you obtain the result in another fashion?

EXERCISE 4.4   Calculate the mean and the variance of the estimate $\hat\beta$ in (3.50).

EXERCISE 4.5   Compute the conditional moments $E( X_{2}\mid x_{1})$ and $E( X_{1}\mid x_{2})$ for the pdf of Example 4.5.

EXERCISE 4.6   Prove the relation (4.28).

EXERCISE 4.7   Prove the relation (4.29). Hint: Note that $\Var(E(X_2\vert X_1)) = \\
E(E(X_2\vert X_1)\, E(X_2^{\top}\vert X_1)) - E(X_2)\, E(X_2^{\top}))$ and that $E(\Var(X_2\vert X_1)) = E[E(X_2X_2^{\top}\vert X_1) - E(X_2\vert X_1) \, E(X_2^{\top}\vert X_1)]$.

EXERCISE 4.8   Compute (4.46) for the pdf of Example 4.5.

EXERCISE 4.9  

\begin{displaymath}\text{Show that}\quad f_Y(y)=
\begin{cases}\frac{1}{2}y_1 -\f...
...rt 1-y_1\vert \\ 0 & otherwise\end{cases} \quad\text{is a pdf!}\end{displaymath}

EXERCISE 4.10   Compute (4.46) for a two-dimensional standard normal distribution. Show that the transformed random variables $Y_{1}$ and $Y_{2}$ are independent. Give a geometrical interpretation of this result based on iso-distance curves.

EXERCISE 4.11   Consider the Cauchy distribution which has no moment, so that the CLT cannot be applied. Simulate the distribution of $\overline{x}$ (for different $n$'s). What can you expect for $n\to\infty$?
Hint: The Cauchy distribution can be simulated by the quotient of two independent standard normally distributed random variables.

EXERCISE 4.12   A European car company has tested a new model and reports the consumption of gasoline ($X_{1})$ and oil ($X_{2}$). The expected consumption of gasoline is 8 liters per 100 km ($\mu_{1}$) and the expected consumption of oil is 1 liter per 10.000 km ($\mu_{2}$). The measured consumption of gasoline is 8.1 liters per 100 km ( $\overline{x}_{1}$) and the measured consumption of oil is 1.1 liters per 10,000 km ( $\overline{x}_{2}$). The asymptotic distribution of $\sqrt{n} \left \{ {\overline{x}_{1} \choose \overline{x}_{2}}
- {\mu_{1} \choose \mu_{2}} \right \}$ is $ N \left (
{0 \choose 0}, \left ( {0.1 \atop 0.05}{0.05 \atop 0.1}
\right ) \right ) $.

For the American market the basic measuring units are miles (1 mile $\approx$ 1.6 km) and gallons (1 gallon $\approx$ 3.8 liter). The consumptions of gasoline ($Y_{1}$) and oil ($Y_{2}$) are usually reported in miles per gallon. Can you express $\overline y_{1}$ and $\overline y_{2}$ in terms of $\overline x_{1}$ and $\overline x_{2}$? Recompute the asymptotic distribution for the American market!

EXERCISE 4.13   Consider the pdf $f(x_1,x_2)=e^{-(x_1+x_2)}, x_1,x_2>0$ and let $U_1=X_1+X_2$ and $U_2=X_1-X_2$. Compute $f(u_1,u_2)$.

EXERCISE 4.14   Consider the pdf`s

\begin{displaymath}
\begin{array}{lclcl}
f(x_1,x_2)&=& 4x_1x_2e^{-x_1^2}&\ &x_1,...
...\frac{1}{2}e^{-x_1}&\ &x_1>\vert x_2\vert.\nonumber
\end{array}\end{displaymath}

For each of these pdf`s compute $E(X), \Var(X), E(X_1\vert X_2), E(X_2\vert X_1), V(X_1\vert X_2)$ and $V(X_2\vert X_1).$

EXERCISE 4.15   Consider the pdf $f(x_1,x_2)=\frac{3}{2}x_1^{-\frac{1}{2}},\ 0<x_1<x_2<1$. Compute $P(X_1<0.25), P(X_2<0.25)$ and $P(X_2<0.25\vert X_1<0.25).$

EXERCISE 4.16   Consider the pdf $f(x_1,x_2)=\frac{1}{2\pi},\ 0<x_1<2\pi, 0<x_2<1.$
Let $U_1=\sin X_1\sqrt{-2\log X_2}$ and $U_2=\cos X_1\sqrt{-2\log X_2}$. Compute $f(u_1,u_2)$.

EXERCISE 4.17   Consider $f(x_1,x_2,x_3)=k(x_1+x_2x_3); \ 0<x_1,x_2,x_3<1.$
a)
Determine $k$ so that $f$ is a valid pdf of $(X_1, X_2, X_3)=X.$
b)
Compute the $(3\times 3)$ matrix $\Sigma_X$.
c)
Compute the $(2\times 2)$ matrix of the conditional variance of $(X_2, X_3)$ given $X_1=x_1$.

EXERCISE 4.18   Let $X \sim N_2 \left({1 \choose 2}, \left(\begin{array}{cc}
2 & a\\
a &2
\end{array}\right)\right)$.
a)
Represent the contour ellipses for $a=0;\ -\frac{1}{2};\ +\frac{1}{2};\ 1.$
b)
For $a=\frac{1}{2}$ find the regions of $X$ centered on $\mu$ which cover the area of the true parameter with probability $0.90$ and $0.95$.

EXERCISE 4.19   Consider the pdf

\begin{displaymath}f(x_1,x_2)=\frac{1}{8x_2}e^{-\left(\frac{x_1}{2x_2}+\frac{x_2}{4}
\right)} \qquad x_1,x_2>0.\end{displaymath}

Compute $f(x_2)$ and $f(x_1\vert x_2)$. Also give the best approximation of $X_1$ by a function of $X_2$. Compute the variance of the error of the approximation.

EXERCISE 4.20   Prove Theorem 4.6.