Homework Assignments

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hw	due	chapter	problems
1	Sep 15	1	2, 4, 5ace, 8abc, 25ab, 26
		2	2, 4ac, 6, 9ac
2	Sep 24	2	12, 17, 20, 26, 27, 31ag, 36ab, 39, 46
3	Oct 1	3	3abc, 4, 6, 12, 14a, 19, 23, 24, 37
		A	1
4	Oct 8	3	27abd, 33, 41, 43, 48a, 52ab, 53b, 61, 70
		A	2
5	Oct 15	4	4, 8, 10, 20, 26b, 29, 32
		A	3, 4, 5
6	Oct 22	4	40ab, 44, 49
		A	6, 7
7	Nov 3	4	55, 58, 73, 74, 76
		5	3, 6, 8, 9
		A	8, 9
8	Nov 10	6	2, 6, 13, 23, 25, 30, 37, 44, 45
		A	10, 11, 15
9	Nov 19	6	48, 49, 50, 58abef, 60, 69, 73
		12	1, 3, 6
		A	12, 13, 14, 16
10	Nov 29	6	82, 85, 92, 93
		12	11, 12, 14
11	Dec 8	12	19, 21
		A	17, 18, 19, 20, 21, 22
12	Dec 15	7	1a-e, 4a-c, 6, 13, 14, 16, 22, 25

Note: Chapter ``A'' refers to the ``additional problems'' below.

Additional Problems

Problem 1   Suppose the random variable X has p. d. f.

$$f(x) = \frac{1}{\sqrt{2 \pi}} e^{- x^2 / 2}, \qquad - \infty < x < + \infty$$

What is the p. d. f. of Y = X³?

Problem 2   For what real values of $\theta$ is

$$f_\theta(x) = \frac{1}{c(\theta)} x^\theta, \qquad 0 < x < 1$$

a probability density, and what is the function $c(\theta)$?

Problem 3   For the densities in Problem 4-8 in Lindgren, find the medians of the distributions.

Problem 4   Show that if Z has mean zero and variance one, then $X = \mu + \sigma Z$ has mean $\mu$ and variance $\sigma^2$(this is the converse to problem 4-32).

Problem 5   Suppose X, Y, and Z are random variables such that

$$E(X \vert Y, Z) = Y \quad \text{and} \quad \mathop{\rm var}(X \vert Y, Z) = Z.$$

Find the (unconditional) mean and variance of X in terms of the means, variances, and covariance of Y and Z.

Problem 6   Suppose X₁, X₂, $\ldots$ are i. i. d. with mean $\mu$ and variance $\sigma^2$. Find the mean and variance of

$$Y_n = \frac{1}{n} \left( X_1 + X_2 + \cdots + X_n \right).$$

Problem 7   Suppose X₁, X₂, $\ldots$, X_n are exchangeable and

$$X_1 + X_2 + \cdots + X_n = 0$$

with probability one. What is the correlation of the X_i?

Problem 8   Suppose that S₁, S₂, $\ldots$ is any sequence of random variables such that $S_n \stackrel{P}{\longrightarrow}\sigma$, and X₁, X₂, $\ldots$ are independent and identically distributed with mean $\mu$ and variance $\sigma^2$. Show that

$$\frac{\overline{X}_n - \mu}{S_n / \sqrt{n}} \stackrel{\mathca... ...rightarrow}\mathcal{N}(0,1), \qquad \text{as $n \to \infty$},$$

where, as usual,

$$\overline{X}_n = \frac{1}{n} \sum_{i=1}^n X_i$$

Problem 9   Suppose X₁, X₂, $\ldots$ are i. i. d. with common probability measure P, and define Y_n = I_A(X_n) for some event A, that is,

$$Y_n = \begin{cases}1, & X_n \in A \\ 0, & X_n \notin A \end{cases}$$

Show that $\overline{Y}_n \stackrel{P}{\longrightarrow}P(A)$.

Problem 10   A gambler makes 100 one-dollar bets on red at roulette. The probability of winning a single bet is 18 / 38. The bets pay even odds, so the gambler gains $1 when he wins and loses $1 when he loses.

What is the mean and the standard deviation of the gambler's net gain (amount won minus amount lost) on the 100 bets?

Problem 11   Suppose X₁, X₂, $\ldots$ are i. i. d. random variables with mean $\mu$ and variance $\sigma^2$, and N is a $\text{Geo}(p)$ random variable independent of the X_i. What is the mean and variance of

$$Y = X_1 + X_2 + \cdots + X_N$$

(note N is random).

Problem 12   A brand of raisin bran averages 84.2 raisins per box. The boxes are filled from large bins of well mixed raisin bran. What is the standard deviation of the number of raisins per box.

Problem 13   Suppose $X \sim \text{Gam}(\alpha, \lambda)$. Let Y = 1 / X.

1. For which values of $\alpha$ and $\lambda$ does E(Y) exist?
2. What is E(Y) when it exists?

Problem 14   Suppose that X, Y, and Z are independent $\mathcal{N}(2, 2)$ random variables. What is P(X > Y + Z)? Hint: What is the distribution of X - Y - Z?

Problem 15   Let X be the number of winners of a lottery. If we assume that players pick their lottery numbers at random, then their choices are i. i. d. random variables and X is binomially distributed. Since the mean number of winners is small, the Poisson approximation is very good. Hence we may assume that $X \sim \text{Poi}(\mu)$ where $\mu$is a constant that depends on the rules of the lottery and the number of tickets sold.

Because of our independence assumption, what other players do is independent of what you do. Hence the conditional distribution of the number of other winners given that you win is also $\text{Poi}(\mu)$. If you are lucky enough to win, you must split the prize with X other winners. You win A / (X + 1) where A is the total prize money. Thus

$$E\left(\frac{A}{X + 1}\right)$$

is your expected winnings given that you win. Calculate this expectation.

Problem 16   Suppose the conditional distribution of Y given X is $\mathcal{N}(0, 1 / X)$ and the marginal distribution of X is $\text{Gam}(\alpha, \lambda)$. What is the marginal p. d. f. of Y?

Problem 17   Suppose X and Z are independent standard normal random variables, and Y = X + X² + Z. Find the following

1. Find the function of X that is the best predictor of Y.
   (Answer: X + X².)
2. Find the mean square prediction error of this predictor.
   (Answer: 1.)
3. Find the function of X that is the best linear predictor of Y.
   (Answer: 1 + X.)
4. Find the mean square prediction error of this predictor.
   (Answer: 3.)

Problem 18   Let M be any symmetric positive semi-definite matrix, and denote its elements m_{i j}. Show that for any i and j

$$- 1 \le \frac{m_{i j}}{\sqrt{m_{i i} m_{j j}}} \le 1$$ (1)

Hint: Consider w' M w for vectors whaving all elements zero except the i-th and j-th.

The point of the problem (this isn't part of the problem, just the explanation of why it is interesting) is that if $\mathbf{M} = \mathop{\rm var}(\mathbf{X})$, then the fraction in (1) is $\mathop{\rm cor}(X_i, X_j)$. Thus positive semi-definiteness is a stronger requirement than the correlation inequality.

Problem 19   Suppose   X₁, $\ldots$, X_n are exchangeable random variables. Show that

$$- \frac{1}{n - 1} \le \mathop{\rm cor}(X_i, X_j).$$

Hint: Consider $\mathop{\rm var}(X_1 + \cdots + X_n)$.

Problem 20   An infinite sequence of random variables X₁, X₂, $\ldots$ is said to be exchangeable if the finite sequence X₁, $\ldots$, X_nis exchangeable for each n.

1. Show that correlations $\mathop{\rm cor}(X_i, X_j)$for an exchangeable infinite sequence must be nonnegative. Hint: Consider Problem 19.
2. Show that the following construction gives an exchangeable infinite sequence X₁, X₂, $\ldots$ of random variables having any correlation in the range $0 \le \rho \le 1$. Let Y₁, Y₂, $\ldots$be an i. i. d. sequence of random variables with variance $\sigma^2$, let Z be a random variable independent of all the Y_i with variance $\tau^2$, and define X_i = Y_i + Z.

Problem 21   If $\mathbf{X} \sim \mathcal{N}(0, \mathbf{M})$ is a nondegenerate normal random vector, what is the distribution of Y = M^-1 X?

Problem 22   Suppose Z₁, Z₂, $\ldots$ are i. i. d. $\mathcal{N}(0, \tau^2)$ random variables and X₁, X₂, $\ldots$ are defined recursively as follows.

• X₁ is a $\mathcal{N}(0, \sigma^2)$ random variable that is independent of all the Z_i.
• for i > 1
  
  $$X_{i + 1} = \rho X_i + Z_i.$$
  
  

There are three unknown parameters, $\rho$, $\sigma^2$, and $\tau^2$, in this model. Because they are variances, we must have $\sigma^2 > 0$and $\tau^2 > 0$. The model is called an autoregressive time series of order one or AR(1) for short. The model is said to be stationary if X_i has the same marginal distribution for all i.

1. Show that the joint distribution of X₁, X₂, $\ldots$, X_n is multivariate normal.
2. Show that E(X_i) = 0 for all i.
3. Show that the model is stationary only if $\rho^2 < 1$ and
   
   $$\sigma^2 = \frac{\tau^2}{1 - \rho^2}$$
   
   

   Hint: Consider $\mathop{\rm var}(X_i)$.

4. Show that
   
   $$\mathop{\rm cov}(X_i, X_{i + k}) = \rho^k \sigma^2, \qquad k \ge 0$$
   
   
   in the stationary model.