Please see the **detailed homework policy document** for information about homework formatting, submission, and grading.

Let \(X_{1}, X_{2}, \ldots X_{n}\) be a random sample of size \(n\) from a distribution with probability density function

\[ f(x, \theta) = \frac{1}{\theta}e^{-x/\theta}, \quad x > 0, \ \theta > 0 \]

Note that, the moments of this distribution are given by

\[ E[X^k] = \int_{0}^{\infty} \frac{x^k}{\theta}e^{-x/\theta} = k! \cdot \theta^k. \]

This will be a useful fact for Exercises 2 and 3.

**(a)** Obtain the maximum likelihood *estimator* of \(\theta\), \(\hat{\theta}\). (This should be a function of the unobserved \(x_i\) and the sample size \(n\).) Calculate the *estimate* when

\[ x_{1} = 0.50, \ x_{2} = 1.50, \ x_{3} = 4.00, \ x_{4} = 3.00. \]

(This should be a single number, for this dataset.)

**(b)** Calculate the bias of the maximum likelihood *estimator* of \(\theta\), \(\hat{\theta}\). (This will be a number.)

**(c)** Find the mean squared error of the maximum likelihood *estimator* of \(\theta\), \(\hat{\theta}\). (This will be an expression based on the parameter \(\theta\) and the sample size \(n\). Be aware of your answer to the previous part, as well as the distribution given.)

**(d)** Provide an *estimate* for \(P[X > 4]\) when

\[ x_{1} = 0.50, \ x_{2} = 1.50, \ x_{3} = 4.00, \ x_{4} = 3.00. \]

Let \(X_{1}, X_{2}, \ldots X_{n}\) be a random sample of size \(n\) from a distribution with probability density function

\[ f(x, \alpha) = \alpha^{-2}xe^{-x/\alpha}, \quad x > 0, \ \alpha > 0 \]

**(a)** Obtain the maximum likelihood *estimator* of \(\alpha\), \(\hat{\alpha}\). Calculate the *estimate* when

\[ x_{1} = 0.25, \ x_{2} = 0.75, \ x_{3} = 1.50, \ x_{4} = 2.5, \ x_{5} = 2.0. \]

**(b)** Obtain the method of moments *estimator* of \(\alpha\), \(\tilde{\alpha}\). Calculate the *estimate* when

\[ x_{1} = 0.25, \ x_{2} = 0.75, \ x_{3} = 1.50, \ x_{4} = 2.5, \ x_{5} = 2.0. \]

Let \(X_{1}, X_{2}, \ldots X_{n}\) be a random sample of size \(n\) from a distribution with probability density function

\[ f(x, \beta) = \frac{1}{2 \beta^3} x^2 e^{-x/\beta}, \quad x > 0, \ \beta > 0 \]

**(a)** Obtain the maximum likelihood *estimator* of \(\beta\), \(\hat{\beta}\). Calculate the *estimate* when

\[ x_{1} = 2.00, \ x_{2} = 4.00, \ x_{3} = 7.50, \ x_{4} = 3.00. \]

**(b)** Obtain the method of moments *estimator* of \(\beta\), \(\tilde{\beta}\). Calculate the *estimate* when

\[ x_{1} = 2.00, \ x_{2} = 4.00, \ x_{3} = 7.50, \ x_{4} = 3.00. \]

\[ f(x, \lambda) = \lambda x^{\lambda - 1}, \quad 0 < x < 1, \lambda > 0 \]

**(a)** Obtain the maximum likelihood *estimator* of \(\lambda\), \(\hat{\lambda}\). Calculate the *estimate* when

\[ x_{1} = 0.10, \ x_{2} = 0.20, \ x_{3} = 0.30, \ x_{4} = 0.40. \]

**(b)** Obtain the method of moments *estimator* of \(\lambda\), \(\tilde{\lambda}\). Calculate the *estimate* when

\[ x_{1} = 0.10, \ x_{2} = 0.20, \ x_{3} = 0.30, \ x_{4} = 0.40. \]