Note: Homework assignments are to be done without using R or any other statistical software. Clear hand written or printed assignments are accepted. Please do not do problems side-by-side.

1 Unbiased vs Biased

Suppose \(X_1, \dots, X_n\) is a random sample from a Poisson distribution with parameter \(\lambda > 0\).

  1. Show that \(\bar{X}\) is an unbiased estimator of \(\lambda\).

  2. Determine whether \(T = \frac{n-1}{n}\bar{X}\) is unbiased or biased for \(\lambda\).

2 Method of Moments

Suppose \(X_1, \dots, X_n\) is a random sample from a distribution with pdf

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

  1. Find the method of moments estimator of \(\theta\).

  2. Determine whether your estimator is unbiased.

3 Method of Moments (Gamma with two parameters)

Suppose \(X_1,\dots,X_n\) is a random sample from a Gamma distribution with shape \(\alpha>0\) and scale \(\beta>0\), written \(X \sim \text{Gamma}(\alpha,\beta)\).

  1. Find method of moments estimators \(\hat\alpha_{\text{MOM}}\) and \(\hat\beta_{\text{MOM}}\).

Probability and Statistical Inference - 9th Edition