# Exercise 1

Consider independent random variables $$X_1$$, $$X_2$$, and $$X_3$$ with

• $$\text{E}[X_1] = 1$$, $$\text{Var}[X_1] = 4$$
• $$\text{E}[X_2] = 2$$, $$\text{SD}[X_1] = 3$$
• $$\text{E}[X_3] = 3$$, $$\text{SD}[X_1] = 5$$

(a) Calculate $$\text{E}[5 X_1 + 2]$$.

(b) Calculate $$\text{E}[4 X_1 + 2 X_2 - 6 X_3]$$.

(c) Calculate $$\text{Var}[5 X_1 + 2]$$.

(d) Calculate $$\text{Var}[4 X_1 + 2 X_2 - 6 X_3]$$.

# Exercise 2

Consider random variables $$H$$ and $$Q$$ with

• $$\text{E}[H] = 3$$, $$\text{Var}[H] = 16$$
• $$\text{SD}[Q] = 4$$, $$\text{E}\left[\frac{Q^2}{5}\right] = 3.2$$

(a) Calculate $$\text{E}[5H^2 - 10]$$.

(b) Calculate $$\text{E}[Q]$$.

# Exercise 3

Consider a random variable $$S$$ with probability density function

$f(s) = \frac{1}{9000}(2s + 10), \ \ 40 \leq s \leq 100.$

(a) Calculate $$\text{E}[S]$$.

(b) Calculate $$\text{SD}[S]$$.

# Exercise 4

Consider independent random variables $$X$$ and $$Y$$ with

• $$X \sim N(\mu_X = 2, \sigma^2_X = 9)$$
• $$Y \sim N(\mu_Y = 5, \sigma^2_Y = 4)$$

(a) Calculate $$P[X > 5]$$.

(b) Calculate $$P[X + 2Y > 5]$$.

# Exercise 5

Consider random variables $$Y_1$$, $$Y_2$$, and $$Y_3$$ with

• $$\text{E}[Y_1] = 1$$, $$\text{E}[Y_2] = -2$$, $$\text{E}[Y_3] = 3$$
• $$\text{Var}[Y_1] = 4$$, $$\text{Var}[Y_2] = 6$$, $$\text{Var}[Y_3] = 8$$
• $$\text{Cov}[Y_1, Y_2] = 1$$, $$\text{Cov}[Y_1, Y_3] = -1$$, $$\text{Cov}[Y_2, Y_3] = 0$$

(a) Calculate $$\text{Var}[3Y_1 - 2Y_2]$$.

(b) Calculate $$\text{Var}[3Y_1 - 4Y_2 + 2Y_3]$$.

# Exercise 6

Consider using $$\hat{\xi}$$ to estimate $$\xi$$.

(a) If $$\text{Bias}[\hat{\xi}] = 5$$ and $$\text{Var}[\hat{\xi}] = 4$$, calculate $$\text{MSE}[\hat{\xi}]$$

(b) If $$\hat{\xi}$$ is unbiased, $$\xi = 6$$, and $$\text{MSE}[\hat{\xi}] = 30$$, calculate $$\text{E}\left[\hat{\xi}^2\right]$$

# Exercise 7

Using the identity

$(\hat{\theta}-\theta) = \left(\hat{\theta}-\text{E}[\hat{\theta}]\right) + \left(\text{E}[\hat{\theta}] - \theta\right) = \left(\hat{\theta} - \text{E}[\hat{\theta}]\right) + \text{Bias}[\hat{\theta}]$

show that

$\text{MSE}[\hat{\theta}] = \text{E}\left[(\hat{\theta} - \theta)^2\right] = \text{Var}[\hat{\theta}] + \left(\text{Bias}[\hat{\theta}]\right)^2$

# Exercise 8

Let $$X_1, X_2, \ldots, X_n$$ denote a random sample from a population with mean $$\mu$$ and variance $$\sigma^2$$.

Consider three estimators of $$\mu$$:

$\hat{\mu}_1 = \frac{X_1 + X_2 + X_3}{3}, ~~~\hat{\mu}_2 = \frac{X_1}{4} + \frac{X_2 + \cdots + X_{n - 1}}{2(n - 2)} + \frac{X_n}{4}, ~~~\hat{\mu}_3 = \bar{X},$

Calculate the mean squared error for each estimator. (It will be useful to first calculate their bias and variances.)

# Exercise 9

Let $$X_1, X_2, \ldots, X_n$$ denote a random sample from a distribution with density

$f(x) = \frac{1}{\theta}e^{-x/\theta}, ~~x > 0, \theta \geq 0$

Consider five estimators of $$\theta$$:

$\hat{\theta}_1 = X_1, ~~~\hat{\theta}_2 = ~~~\frac{X_1 + X_2}{2}, ~~~\hat{\theta}_3 = ~~~\frac{X_1 + 2X_2}{3}, ~~~\hat{\theta}_4 = \bar{X}, ~~~\hat{\theta}_5 = 5$

Calculate the mean squared error for each estimator. (It will be useful to first calculate their bias and variances.)

# Exercise 10

Suppose that $$\text{E}\left[\hat{\theta}_1\right] = \text{E}\left[\hat{\theta}_2\right] = \theta$$, $$\text{Var}\left[\hat{\theta}_1\right] = \sigma_1^2$$, $$\text{Var}\left[\hat{\theta}_2\right] = \sigma_2^2$$, and $$\text{Cov}\left[\hat{\theta}_1, \hat{\theta}_2\right] = \sigma_{12}$$. Consider the unbiased estimator

$\hat{\theta}_3 = a\hat{\theta}_1 + (1-a)\hat{\theta}_2.$

If $$\hat{\theta}_1$$ and $$\hat{\theta}_2$$ are independent, what value should be chosen for the constant $$a$$ in order to minimize the variance and thus mean squared error of $$\hat{\theta}_3$$ as an estimator of $$\theta$$?

# Exercise 11

Let $$Y$$ have a binomial distribution with parameters $$n$$ and $$p$$. Consider two estimators for $$p$$:

$\hat{p}_1 = \frac{Y}{n}$

and

$\hat{p}_2 = \frac{Y + 1}{n + 2}$

For what values of $$p$$ does $$\hat{p}_2$$ achieve a lower mean square error than $$\hat{p}_1$$?

# Exercise 12

Let $$X_1, X_2, \ldots, X_n$$ denote a random sample from a population with mean $$\mu$$ and variance $$\sigma^2$$.

Create an unbiased estimator for $$\mu^2$$. Hint: Start with $$\bar{X}^2$$.

# Exercise 13

Let $$X_1, X_2, X_3, \ldots, X_n$$ be iid random variables form $$\text{U}(\theta, \theta + 2)$$. (That is, a uniform distribution with a minimum of $$\theta$$ and a maximum of $$\theta + 2$$.)

Consider the estimator

$\hat{\theta} = \frac{1}{n}\sum_{i = 1}^{n}X_i = \bar{X}$

(a) Calculate the bias of $$\hat{\theta}$$ when estimating $$\theta$$.

(b) Calculate the variance of $$\hat{\theta}$$.

(c) Calculate the mean squared error of $$\hat{\theta}$$ when estimating $$\theta$$.