Let \(X_1, X_2, \ldots, X_n \stackrel{iid}{\sim} \text{Poisson}(\lambda)\). That is

\[ f(x \mid \lambda) = \frac{\lambda^xe^{-\lambda}}{x!}, \quad x = 0, 1, 2, \ldots \ \ \lambda > 0 \]

**(a)** Obtain a method of moments **estimator** for \(\lambda\), \(\tilde{\lambda}\). Calculate an **estimate** using this *estimator* when

\[ x_{1} = 1, \ x_{2} = 2, \ x_{3} = 4, \ x_{4} = 2. \]

**(b)** Find the maximum likelihood **estimator** for \(\lambda\), \(\hat{\lambda}\). Calculate an **estimate** using this *estimator* when

\[ x_{1} = 1, \ x_{2} = 2, \ x_{3} = 4, \ x_{4} = 2. \]

**(c)** Find the maximum likelihood **estimator** of \(P[X = 4]\), call it \(\hat{P}[X = 4]\). Calculate an **estimate** using this *estimator* when

\[ x_{1} = 1, \ x_{2} = 2, \ x_{3} = 4, \ x_{4} = 2. \]

Let \(X_1, X_2, \ldots, X_n \stackrel{iid}{\sim} N(\theta,\sigma^2)\).

Find a method of moments **estimator** for the *parameter vector* \(\left(\theta, \sigma^2\right)\).

Let \(X_1, X_2, \ldots, X_n \stackrel{iid}{\sim} N(1,\sigma^2)\).

Find a method of moments **estimator** of \(\sigma^2\), call it \(\tilde{\sigma}^2\).

Let \(X_1, X_2, \ldots, X_n\) be a random sample from a population with pdf

\[ f(x \mid \theta) = \frac{1}{\theta}x^{(1-\theta)/\theta}, \quad 0 < x < 1, \ 0 < \theta < \infty \]

**(a)** Find the maximum likelihood **estimator** of \(\theta\), call it \(\hat{\theta}\). Calculate an **estimate** using this *estimator* when

\[ x_{1} = 0.10, \ x_{2} = 0.22, \ x_{3} = 0.54, \ x_{4} = 0.36. \]

**(b)** Obtain a method of moments **estimator** for \(\theta\), \(\tilde{\theta}\). Calculate an **estimate** using this *estimator* when

\[ x_{1} = 0.10, \ x_{2} = 0.22, \ x_{3} = 0.54, \ x_{4} = 0.36. \]

Let \(X_1, X_2, \ldots, X_n\) iid from a population with pdf

\[ f(x \mid \theta) = \frac{\theta}{x^2}, \quad 0 < \theta \leq x \]

Obtain the maximum likelihood **estimator** for \(\theta\), \(\hat{\theta}\).

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

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

**Hint:** Recall the probability density function of an exponential random variable.

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

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 hint will also be useful in the next exercise.

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

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

**(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

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

Let \(Y_1, Y_2, \ldots, Y_n\) be a random sample from a distribution with pdf

\[ f(y \mid \alpha) = \frac{2}{\alpha} \cdot y \cdot \exp\left\{-\frac{y^2}{\alpha}\right\}, \ y > 0, \ \alpha > 0. \]

**(a)** Find the maximum likelihood **estimator** of \(\alpha\).

**(b)** Let \(Z_1 = Y_1^2\). Find the distribution of \(Z_1\). Is the MLE for \(\alpha\) an unbiased estimator of \(\alpha\)?

Let \(X\) be a single observation from a \(\text{Binom}(n, p),\) where \(p\) is an unknown parameter. (In this case, we will consider \(n\) known.)

**(a)** Find the maximum likelihood **estimator** (MLE) of \(p\).

**(b)** Suppose you roll a 6-sided die 40 times and observe eight rolls of a 6. What is the maximum likelihood **estimate** of the probability of observing a 6?

**(c)** Using the same observed data, suppose you now plan to perform a second experiment with the same die, and will roll the die 5 more times. What is the maximum likelihood **estimate** of the probability that you will observe no 6’s in this next experiment?

Suppose that a random variable \(X\) follows a discrete distribution, which is determined by a parameter \(\theta\) which can take *only two values*, \(\theta = 1\) or \(\theta = 2\). The parameter \(\theta\) is unknown.

- If \(\theta = 1\), then \(X\) follows a Poisson distribution with parameter \(\lambda = 2\).
- If \(\theta = 2\), then \(X\) follows a Geometric distribution with parameter \(p = \frac{1}{4}\).

Now suppose we observe \(X = 3\). Based on this data, what is the maximum likelihood **estimate** of \(\theta\)?

Let \(Y_1, Y_2, \ldots, Y_n\) be a random sample from a population with pdf

\[ f(y \mid \theta) = \dfrac{2\theta^2}{y^3}, \ \ \theta \le y < \infty \]

Find the maximum likelihood **estimator** of \(\theta.\).