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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 14 hours.

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 \(\theta\)?

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

\[ \bar{X}. \]

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.

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 \(C = 5Y + Y^2\).

Given that \(\text{E}[\bar{Y}] = \lambda\) and \(\text{E}[C] = 6\lambda + \lambda^2\), 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 \(a\bar{Y} + b\bar{Y}^2\).

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?