23.5 Exercises

1. A random vector $\mathbf{Y} = [Y_1 ~~ Y_2 ~~ \cdots ~~ Y_n]^T$ has mean vector $\boldsymbol{\mu}$ and covariance matrix $\sigma^2 \mathbf{I}_n$ where $\sigma > 0$ is a number and $\mathbf{I}_n$ is the $n \times n$ identity matrix.

(a) Pick one option and explain: $Y_1$ and $Y_2$ are

$~~~~~$ (i) independent. $~~~~~~~~$ (ii) uncorrelated but might not be independent. $~~~~~~~~$ (iii) not uncorrelated.

(b) Pick one option and explain: $Var(Y_1)$ and $Var(Y_2)$ are

$~~~~~$ (i) equal. $~~~~~~~~$ (ii) possibly equal, but might not be. $~~~~~~~~$ (iii) not equal.

(c) For $m \le n$ let $\mathbf{A}$ be an $m \times n$ matrix of real numbers, and let $\mathbf{b}$ be an $m \times 1$ vector of real numbers. Let $\mathbf{V} = \mathbf{AY} + \mathbf{b}$. Find the mean vector $\boldsymbol{\mu}_\mathbf{V}$ and covariance matrix $\boldsymbol{\Sigma}_\mathbf{V}$ of $\mathbf{V}$.

(d) Let $\mathbf{c}$ be an $m \times 1$ vector of real numbers and let $W = \mathbf{c}^T\mathbf{V}$ for $\mathbf{V}$ defined in Part (c). In terms of $\mathbf{c}$, $\boldsymbol{\mu}_\mathbf{V}$ and $\boldsymbol{\Sigma}_\mathbf{V}$, find $E(W)$ and $Var(W)$.

2. Let $[U ~ V ~ W]^T$ be multivariate normal with mean vector $[0 ~ 0 ~ 0]^T$ and covariance matrix $\begin{bmatrix} 1 & \rho_1 & \rho_2 \\ \rho_1 & 1 & \rho_3 \\ \rho_2 & \rho_3 & 1 \end{bmatrix}$

(a) What is the distribution of $U$?

(b) What is the distribution of $U+2V$?

(c) What is the joint distribution of $U$ and $U+2V$?

(d) Under what condition on the parameters is $U$ independent of $U+2V$?

3. Let $[X_1 ~~ X_2 ~~ X_3]^T$ be multivariate normal with mean vector $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$ given by

Find $P\big( (X_1 + X_2)/2 < X_3 + 1 \big)$.

4. Let $X$ be standard normal. Construct a random variable $Y$ as follows:

• Toss a fair coin.

• If the coin lands heads, let $Y = X$.

• If the coin lands tails, let $Y = -X$.

(a) Find the cdf of $Y$ and hence identify the distribution of $Y$.

(b) Find $E(XY)$ by conditioning on the result of the toss.

(c) Are $X$ and $Y$ uncorrelated?

(d) Are $X$ and $Y$ independent?

(e) Is the joint distribution of $X$ and $Y$ bivariate normal?

5. Normal Sample Mean and Sample Variance, Part 1

Let $X_1, X_2, \ldots, X_n$ be i.i.d. with mean $\mu$ and variance $\sigma^2$. Let

denote the sample mean and

denote the sample variance as defined earlier in the course.

(a) For $1 \le i \le n$ let $D_i = X_i - \bar{X}$. Find $Cov(D_i, \bar{X})$.

(b) Now assume in addition that $X_1, X_2, \ldots, X_n$ are i.i.d. normal $(\mu, \sigma^2)$. What is the joint distribution of $\bar{X}, D_1, D_2, \ldots, D_{n-1}$? Explain why $D_n$ isn’t on the list.

(c) True or false (justify your answer): The sample mean and sample variance of an i.i.d. normal sample are independent of each other.

6. Normal Sample Mean and Sample Variance, Part 2

(a) Let $R$ have the chi-squared distribution with $n$ degrees of freedom. What is the mgf of $R$?

(b) For $R$ as in Part (a), suppose $R = V + W$ where $V$ and $W$ are independent and $V$ has the chi-squared distribution with $m < n$ degrees of freedom. Can you identify the distribution of $W$? Justify your answer.

(c) Let $X_1, X_2, \ldots , X_n$ be any sequence of random variables and let $\bar{X} = \frac{1}{n}\sum_{i=1}^n X_i$. Let $\alpha$ be any constant. Prove the sum of squares decomposition

(d) Now let $X_1, X_2, \ldots, X_n$ be i.i.d. normal with mean $\mu$ and variance $\sigma^2 > 0$. Let $S^2$ be the “sample variance” defined by

Find a constant $c$ such that $cS^2$ has a chi-squared distribution. Provide the degrees of freedom.

[Use Parts (b) and (c) as well as the result of the previous exercise.]