The Multivariate Normal Distribution

In this chapter we have (so far) met three equivalent definitions of a multivariate normal distribution. Here is a fourth one: $\mathbf{X}$ is normal if and only if there exists an orthogonal transformation $\mathbf{C}$ such that the random vector $\mathbf{CX}$ has independent, normal components. Show that this definition is indeed equivalent to the usual ones (e.g., by showing that it is equivalent to the first one).

Suppose that $X$ and $Y$ have a two-dimensional normal distribution with means $0$, variances $1$, and correlation coefficient $\rho$, $\left|\rho \right| < 1$. Let $(R, \Theta )$ be the polar coordinates. Determine the distribution of $\Theta$.

The random variables $X_{1}$ and $X_{2}$ are independent and $N(0,1)$-distributed. Set

Show that $Y_{1}$ and $Y_{2}$ are independent, $N(0,1)$-distributed random variables.

The random vector $(X, Y)'$ has a two-dimensional normal distribution with $\operatorname {Var}\left[X\right] = \operatorname {Var}\left[Y\right]$. Show that $X+Y$ and $X-Y$ are independent random variables.

Suppose that $X$ and $Y$ have a joint normal distribution with $\mathbb {E}\left[X\right] = \mathbb {E}\left[Y\right] = 0$, $\operatorname {Var}\left[X\right] = \sigma_{x}^{2}$, $\operatorname {Var}\left[Y\right] = \sigma_{y}^{2}$, and correlation coefficient $\rho$. Compute $\mathbb {E}\left[XY\right]$ and $\operatorname {Var}\left[XY\right]$.

The random variables $X$ and $Y$ are independent and $N(0,1)$-distributed. Determine

We know from Section 7 that if $X$ and $Y$ are jointly normally distributed then they are independent iff they are uncorrelated. Now, let $X \in N(0,1)$ and $c \geq 0$. Define $Y$ as follows:

In Section 6 we found that conditional distributions of normal vectors are normal. The converse is, however, not true. Namely, consider the bivariate density

where $C$ is a normalizing constant. This is not a bivariate normal density. Show that in spite of this the conditional distributions are normal, that is, compute the conditional densities $f_{Y \mid X=x}(y)$ and $f_{X \mid Y=y}(x)$ and show that they are normal densities.

Suppose that the random variables $X$ and $Y$ are independent and $N(0,\sigma^{2})$-distributed.

Suppose that the moment generating function of $(X,Y)'$ is

Determine $a$ so that $X+2Y$ and $2X-Y$ become independent.

Let $\mathbf{X}$ have a three-dimensional normal distribution. Show that if $X_{1}$ and $X_{2}+X_{3}$ are independent, $X_{2}$ and $X_{1}+X_{3}$ are independent, and $X_{3}$ and $X_{1}+X_{2}$ are independent, then $X_{1}$, $X_{2}$, and $X_{3}$ are independent.

Let $X_{1}$ and $X_{2}$ be independent, $N(0,1)$-distributed random variables. Set $Y_{1} = X_{1}-3X_{2}+2$ and $Y_{2} = 2X_{1}-X_{2}-1$. Determine the distribution of

Let $X_{1}$, $X_{2}$, and $X_{3}$ be independent, $N(1,1)$-distributed random variables. Set $U = 2X_{1}-X_{2}+X_{3}$ and $V = X_{1}+2X_{2}+3X_{3}$. Determine the conditional distribution of $V$ given that $U = 3$.

Let $X_{1}, X_{2}, X_{3}$ be independent $N(2,1)$-distributed random variables. Determine the distribution of $X_{1}+3X_{2}-2X_{3}$ given that $2X_{1}-X_{2} = 1$.

Let $Y_{1}, Y_{2}$, and $Y_{3}$ be independent, $N(0,1)$-distributed random variables, and set

Determine the conditional distribution of $X_{2}$ given that $X_{1}+X_{3} = x$.

The random variables $X_{1}$, $X_{2}$, and $X_{3}$ are independent and $N(0,1)$-distributed. Consider the random variables

Determine the conditional density of $Y_{1}$ given that $Y_{2} = Y_{3} = 0$.

The random vector $\mathbf{X}$ has a three-dimensional normal distribution with mean vector $\mathbf{0}$ and covariance matrix $\mathbf{\Lambda }$ given by

Find the distribution of $X_{2}$ given that $X_{1}-X_{3} = 1$ and that $X_{2}+X_{3} = 0$.

The random vector $\mathbf{X}$ has a three-dimensional normal distribution with expectation $\mathbf{0}$ and covariance matrix $\mathbf{\Lambda }$ given by

Find the distribution of $X_{3}$ given that $X_{1} = 1$.

The random vector $\mathbf{X}$ has a three-dimensional normal distribution with expectation $\mathbf{0}$ and covariance matrix $\mathbf{\Lambda }$ given by

Find the distribution of $X_{2}$ given that $X_{1}+X_{3} = 1$.

The random vector $\mathbf{X}$ has a three-dimensional normal distribution with mean vector $\boldsymbol {\mu } = \mathbf{0}$ and covariance matrix

Find the distribution of $X_{1}+X_{3}$ given that

Let $\mathbf{X} \in N(\boldsymbol {\mu }, \mathbf{\Lambda })$, where

Determine the conditional distribution of $X_{1}-X_{3}$ given that $X_{2} = -1$.

Let $\mathbf{X} \in N(\boldsymbol {\mu }, \mathbf{\Lambda })$, where

Determine the conditional distribution of $X_{1}+X_{2}$ given that $X_{3} = 1$.

The random vector $\mathbf{X}$ has a three-dimensional normal distribution with expectation $\boldsymbol {\mu }$ and covariance matrix $\mathbf{\Lambda }$ given by

Find the conditional distribution of $X_{1}+2X_{2}$ given that

The random vector $\mathbf{X}$ has a three-dimensional normal distribution with mean vector $\boldsymbol {\mu }$ and covariance matrix $\mathbf{\Lambda }$ given by

Find the conditional distribution of $X_{1}$ given that $X_{1} = -X_{2}$.

Let $\mathbf{X}$ have a three-dimensional normal distribution with mean vector and covariance matrix

respectively. Set $Y_{1} = X_{1}+X_{2}+X_{3}$ and $Y_{2} = X_{1}+X_{3}$. Determine the conditional distribution of $Y_{1}$ given that $Y_{2} = 0$.

Let $\mathbf{X} \in N(\mathbf{0}, \mathbf{\Lambda })$, where

Find the conditional distribution of $X_{1}$ given that $X_{1} = X_{2}$ and $X_{1}+X_{2}+X_{3} = 0$.

The random vector $\mathbf{X}$ has a three-dimensional normal distribution with expectation $\mathbf{0}$ and covariance matrix $\mathbf{\Lambda }$ given by

Find the distribution of $X_{2}$ given that $X_{1} = X_{2} = X_{3}$.

Let $\mathbf{X} \in N(\mathbf{0}, \mathbf{\Lambda })$, where

Determine the conditional distribution of $(X_{1}, X_{1}+X_{2})'$ given that $X_{1}+X_{2}+X_{3} = 0$.

Suppose that the characteristic function of $(X,Y,Z)'$ is

Compute the conditional distribution of $X+Z$ given that $X+Y = 0$.

Let $X_{1}$, $X_{2}$, and $X_{3}$ have a joint moment generating function as follows:

Determine the conditional distribution of $X_{1}+X_{3}$ given that $X_{1}+X_{2} = 1$.

The moment generating function of $(X,Y,Z)'$ is

Determine the conditional distribution of $X$ given that $X+Z = 0$ and $Y+Z = 1$.

Suppose $(X,Y,Z)'$ is normal with density

where $C$ is a normalizing constant. Determine the conditional distribution of $X$ given that $X+Z = 1$ and $Y+Z = 0$.

Let $X$ and $Y$ be random variables, such that

Let $X$ and $Y$ be jointly normal with means $0$, variances $1$, and correlation coefficient $\rho$. Compute the moment generating function of $X \cdot Y$ for

Suppose $X_{1}$, $X_{2}$, and $X_{3}$ are independent and $N(0,1)$-distributed. Compute the moment generating function of $Y = X_{1}X_{2}+X_{1}X_{3}+X_{2}X_{3}$.

If $X$ and $Y$ are independent, $N(0,1)$-distributed random variables, then $X^{2}+Y^{2} \in \chi^{2}(2)$ (recall Exercise 3.3.6). Now, let $X$ and $Y$ be jointly normal with means $0$, variances $1$, and correlation coefficient $\rho$. In this case $X^{2}+Y^{2}$ has a noncentral $\chi^{2}(2)$-distribution. Determine the moment generating function of that distribution.

Let $(X,Y)'$ have a two-dimensional normal distribution with means $0$, variances $1$, and correlation coefficient $\rho$, $\left|\rho \right| < 1$. Determine the distribution of $(X^{2}-2\rho XY+Y^{2})/(1-\rho^{2})$ by computing its moment generating function.

Let $X_{1}, X_{2}, \ldots , X_{n}$ be independent, $N(0,1)$-distributed random variables, and set $\overline{X}_{k} = \dfrac {1}{k-1}\sum_{i=1}^{k-1} X_{i}$, $2 \leq k \leq n$. Show that

is $\chi^{2}$-distributed. What is the number of degrees of freedom?

Let $X_{1}$, $X_{2}$, and $X_{3}$ be independent, $N(1,1)$-distributed random variables. Set $U = X_{1}+X_{2}+X_{3}$ and $V = X_{1}+2X_{2}+3X_{3}$. Determine the constants $a$ and $b$ so that $\mathbb {E}\left[(U-a-bV)^{2}\right]$ is minimized.

Let $X$ and $Y$ be independent, $N(0,1)$-distributed random variables. Then $X+Y$ and $X-Y$ are independent; see Example 7.1. The purpose of this problem is to point out a (partial) converse. Suppose that $X$ and $Y$ are independent random variables with common distribution function $F$. Suppose, further, that $F$ is symmetric and that $\sigma^{2} = \mathbb {E}\left[X^{2}\right] < \infty$. Let $\varphi$ be the characteristic function of $X$ (and $Y$). Show that if $X+Y$ and $X-Y$ are independent then we have

Use this relation to show that $\varphi (t) = e^{-\sigma^{2}t^{2}/2}$. Finally, conclude that $F$ is the distribution function of a normal distribution $(N(0,\sigma^{2}))$.