Multivariate Normal Distribution

Suppose that $\varepsilon \sim N_{3}\left(0, \sigma^{2} \mathrm{I}_{3}\right)$ and that $Y_{0}$ is $N\left(0, \sigma_{0}^{2}\right)$, independently of the $\varepsilon_{i}$ 's. Define

Let $\mathbf{Y} \sim N_{n}\left(\mathbf{0}, \mathbf{I}_{n}\right)$, and put $\mathbf{X} = \mathbf{A Y}, \mathbf{U} = \mathbf{B Y}$ and $\mathbf{V} = \mathbf{C Y}$. Suppose that $\operatorname {Cov}\left[ \mathbf{X}, \mathbf{U} \right] = \mathbf{0}$ and $\operatorname {Cov}\left[ \mathbf{X}, \mathbf{V} \right] = \mathbf{0}$. Show that $\mathbf{X}$ is independent of $\mathbf{U}+\mathrm{V}$.

If $Y_{1}, Y_{2}, \ldots , Y_{n}$ is a random sample from $N\left(\mu , \sigma^{2}\right)$, prove that $\bar{Y}$ is independent of $\sum_{i = 1}^{n-1}\left(Y_{i}-Y_{i+1}\right)^{2}$.

If $\mathbf{X}$ and $\mathbf{Y}$ are $n$-dimensional vectors with independent multivariate normal distributions, prove that $a \mathbf{X}+b \mathbf{Y}$ is also multivariate normal.

If $\mathbf{Y} \sim N_{n}\left(\mathbf{0}, \mathbf{I}_{n}\right)$ and $\mathbf{a}$ is a nonzero vector, show that the conditional distribution of $\mathbf{Y}^{T} \mathbf{Y}$ given $\mathbf{a}^{T} \mathbf{Y} = 0$ is $\chi_{n-1}^{2}$.

Let $\mathbf{Y} \sim N_{n}\left(\mu \mathbf{1}_{n}, \mathbf{\Sigma }\right)$, where $\mathbf{\Sigma } = (1-\rho ) \mathbf{I}_{n}+\rho \mathbf{1}_{n} \mathbf{1}_{n}^{T}$ and $\rho > -1 /(n-1)$. Show that $\sum_{i}\left(Y_{i}-\bar{Y}\right)^{2} /(1-\rho )$ is $\chi_{n-1}^{2}$.

Let $\mathbf{Y}_{i}, i = 1, \ldots , n$, be independent $N_{p}(\mathbf{\mu }, \mathbf{\Sigma })$ random vectors. Show that

is an unbiased estimate of $\mathbf{\Sigma }$.

Let $\mathbf{Y} \sim N_{n}\left(\mathbf{0}, \mathbf{I}_{n}\right)$ and let $\mathbf{A}$ and $\mathbf{B}$ be symmetric idempotent matrices with $\mathbf{A B} = \mathbf{B A} = \mathbf{0}$. Show that $\mathbf{Y}^{T} \mathbf{A Y}, \mathbf{Y}^{T} \mathbf{B Y}$ and $\mathbf{Y}^{T}\left(\mathbf{I}_{n}-\mathbf{A}-\mathbf{B}\right) \mathbf{Y}$ have independent chi-square distributions.

Let $\left(X_{i}, Y_{i}\right), i = 1,2, \ldots , n$, be a random sample from a bivariate normal distribution, with means $\mu_{1}$ and $\mu_{2}$, variances $\sigma_{1}^{2}$ and $\sigma_{2}^{2}$, and correlation $\rho$, and let

If $\mathbf{Y} \sim N_{2}(\mathbf{0}, \mathbf{\Sigma })$, where $\mathbf{\Sigma } = \left(\sigma_{i j}\right)$, prove that

Let $a_{0}, a_{1}, \ldots , a_{n}$ be independent $N\left(0, \sigma^{2}\right)$ random variables and define

Show that $\mathbf{Y} = \left(Y_{1}, Y_{2}, \ldots , Y_{n}\right)^{T}$ has a multivariate normal distribution and find its variance-covariance matrix. (The sequence $Y_{1}, Y_{2}, \ldots$ is called a moving average process of order one and is a commonly used model in time series analysis.)

Suppose that $\mathbf{Y} \sim N_{3}\left(\mathbf{0}, \mathbf{I}_{n}\right)$. Find the m.g.f. of $2\left(Y_{1} Y_{2}-Y_{2} Y_{3}-Y_{3} Y_{1}\right)$. Hence show that this random variable has the same distribution as that of $2 U_{1}-U_{2}-U_{3}$, where the $U_{i}$ 's are independent $\chi_{1}^{2}$ random variables.

Theorem 2.3 can be used as a definition of the multivariate normal distribution. If so, deduce from this definition the following results:

Let $\mathbf{Y} = \left(Y_{1}, Y_{2}, \ldots , Y_{n}\right)^{T}$ be a vector of $n$ random variables $(n > 3)$ with density function

Prove that any subset of $n-1$ random variables are mutually independent $N(0,1)$ variables.

Suppose that $\mathbf{Y} = \left(Y_{1}, Y_{2}, Y_{3}, Y_{4}\right)^{T} \sim N_{4}\left(\mathbf{0}, \mathrm{I}_{4}\right)$, and let $Q = Y_{1} Y_{2}-Y_{3} Y_{4}$.

If $\mathbf{Y} \sim N_{n}\left(\mathbf{0}, \mathbf{I}_{n}\right)$, find the variance of

Given $\mathbf{Y} \sim N_{n}(\mathbf{\mu }, \mathbf{\Sigma })$, prove that

Show that

is the density of a bivariate normal random vector $\mathbf{Y} = \left(Y_{1}, Y_{2}\right)^{T}$.

Let $\mathbf{U}$ have density $g$ and let $\mathbf{Y} = \mathbf{A}(\mathbf{U}+\mathbf{c})$, where $\mathbf{A}$ is nonsingular. Show that the density $f$ of $\mathbf{Y}$ satisfies

where $\mathbf{y} = \mathbf{A}(\mathbf{u}+\mathbf{c})$.

Find the moment generating function of the bivariate normal distribution given in Example 2.2.

If $\mathbf{Y} \sim N_{n}(\mathbf{\mu }, \mathbf{\Sigma })$, show that $Y_{i} \sim N\left(\mu_{i}, \sigma_{i i}\right)$.

Suppose that $\mathbf{Y} \sim N_{3}(\mathbf{\mu }, \mathbf{\Sigma })$, where

Find the joint distribution of $Z_{1} = Y_{1}+Y_{2}+Y_{3}$ and $Z_{2} = Y_{1}-Y_{2}$.

Given $\mathbf{Y} \sim N_{n}\left(\mathbf{\mu }, \mathbf{I}_{n}\right)$, find the joint density of $\mathbf{a}^{T} \mathbf{Y}$ and $\mathbf{b}^{T} \mathbf{Y}$, where $\mathbf{a}^{T} \mathbf{b} = 0$, and hence show that $\mathbf{a}^{T} \mathbf{Y}$ and $\mathbf{b}^{T} \mathbf{Y}$ are independent.

Let $\left(X_{i}, Y_{i}\right), i = 1,2, \ldots , n$, be a random sample from a bivariate normal distribution. Find the joint distribution of $(\bar{X}, \bar{Y})$.

If $Y_{1}$ and $Y_{2}$ are random variables such that $Y_{1}+Y_{2}$ and $Y_{1}-Y_{2}$ are independent $N(0,1)$ random variables, show that $Y_{1}$ and $Y_{2}$ have a bivariate normal distribution. Find the mean and variance matrix of $\mathbf{Y} = \left(Y_{1}, Y_{2}\right)^{T}$.

Let $X_{1}$ and $X_{2}$ have joint density

Show that $X_{1}$ and $X_{2}$ have $N(0,1)$ marginal distributions.

Suppose that $Y_{1}, Y_{2}, \ldots , Y_{n}$ are independently distributed as $N(0,1)$. Calculate the m.g.f: of the random vector

and hence show that $\bar{Y}$ is independent of $\sum_{i}\left(Y_{i}-\bar{Y}\right)^{2}$.

Let $X_{1}, X_{2}$, and $X_{3}$ be i.i.d. $N(0,1)$. Let

Show that $Y_{1}, Y_{2}$ and $Y_{3}$ are i.i.d. $N(0,1)$. (The transformation above is a special case of the so-called Helmert transformation.)

If $Y_{1}, Y_{2}, \ldots , Y_{n}$ have a multivariate normal distribution and are pairwise independent, are they mutually independent?

Let $\mathbf{Y} \sim N_{n}\left(\mu \mathbf{1}_{n}, \mathbf{\Sigma }\right)$, where $\mathbf{\Sigma } = (1-\rho ) \mathbf{I}_{n}+\rho \mathbf{J}_{n}$ and $\rho > -1 /(n-1)$. When $\rho = 0, \bar{Y}$ and $\sum_{i}\left(Y_{i}-\bar{Y}\right)^{2}$ are independent, by Example 2.8. Are they independent when $\rho \neq 0$?

Given $\mathbf{Y} \sim N_{3}(\mathbf{\mu }, \mathbf{\Sigma })$, where

for what value(s) of $\rho$ are $Y_{1}+Y_{2}+Y_{3}$ and $Y_{1}-Y_{2}-Y_{3}$ statistically independent?

Show that the m.g.f. for (2.10) is $\prod_{1}^{n}\left(1-2 t d_{i}\right)^{-1 / 2}$.

Let $\mathbf{Y} \sim N_{n}\left(\mathbf{0}, \mathrm{I}_{n}\right)$ and let $\mathbf{A}$ be symmetric.

If $\mathbf{Y} \sim N_{2}\left(0, \mathbf{I}_{2}\right)$, find values of $a$ and $b$ such that

Suppose that $\mathbf{Y} \sim N_{3}\left(\mathbf{0}, \mathbf{I}_{n}\right)$. Show that

has a $\chi_{2}^{2}$ distribution. Does some multiple of

have a chi-squared distribution for general $n$?

Let $\mathbf{Y} \sim N_{n}\left(0, \mathrm{I}_{n}\right)$ and let $\mathbf{A}$ and $\mathbf{B}$ be symmetric. Show that the joint m.g.f. of $\mathbf{Y}^{T} \mathbf{A Y}$ and $\mathbf{Y}^{T} \mathbf{B Y}$ is $\left[\operatorname {det}\left(\mathbf{I}_{n}-2 s \mathbf{A}-2 t B\right)\right]^{-1 / 2}$. Hence show that the two quadratic forms are independent if $\mathbf{A B} = 0$.