Exercise 1

The Dormouse, Garfield, and Snorlax are three notorious sleepers. Since none of them exist in the same fictional universe, each sleeper’s sleep is independent of the others.

• The Dormouse’s sleep follows a normal distribution with a mean of 10 hours and a standard deviation of 2 hours.
• Garfield’s sleep follows a normal distribution with a mean of 12 hours and a standard deviation of 2 hours.
• Snorlax’s sleep follows a normal distribution with a mean of 14 hours and a standard deviation of 1 hour.

Calculate the probability that on some randomly chosen night, this trio’s sleep averages more than 15 hours.

Exercise 2

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

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

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 3

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

$f(x) = \frac{3x^2}{\beta^3}, 0 < x < \beta.$

In order to estimate $$\beta$$, consider the estimator

$\frac{4}{3}\bar{X}.$

Calculate the mean squared error of this estimator.

Hint: You will first need to calculate the expected value and variance of $$X$$. Then calculate the bias and variance of the proposed estimator.

Exercise 4

Suppose that the number of accidents per week for a particular brand of electric scooters follows a Poisson distribution with mean $$\lambda$$. A random sample, $$Y_1, Y_2, \ldots, Y_n$$ of observations on the weekly number of accidents is available. The medical costs for these accidents (in \$1,000s of dollars) is $$C = 4Y + Y^2$$.

Given that $$\text{E}[\bar{Y}] = \lambda$$ and $$\text{E}[C] = 5\lambda + \lambda^2$$, find a function of $$Y_1, Y_2, \ldots, Y_n$$ that is an unbiased estimator for $$\text{E}[C]$$.

Hint: This estimator will be of the form $$a\bar{Y} + b\bar{Y}^2$$.

Exercise 5

Suppose that $$X_1$$, $$X_2$$, $$X_3$$ denote a random sample from a normal distribution with an unknown mean $$\mu$$ and a variance of 1. That is,

$X_i \sim N(\mu, \sigma^2_1 = 1).$

Consider two estimators,

$\hat{\mu}_1 = \frac{1}{3}X_1 + \frac{1}{3}X_2 + \frac{1}{3}X_3$

and

$\hat{\mu}_2 = \frac{1}{9}X_1 + \frac{1}{9}X_2 + \frac{1}{9}X_3.$

For what values of $$\mu$$ does $$\hat{\mu}_2$$ obtain a lower MSE than $$\hat{\mu}_1$$, if any?