Homework 4

Homework 4 is due 04/12/2026 at 11:59 PM. Submit your homework on Canvas as one PDF document.

1. Let \(X_1, X_2, \ldots, X_n\) be iid with the following density function

   \[ f(x) = \left\{\begin{array}{cc} \frac{\alpha x^{\alpha-1}}{3^\alpha}& 0\le x\le 3 \\ 0 & \mathrm{elsewhere} \end{array} \right. \]

   Find the method of moment estimator for \(\alpha\).

2. Let \(X_1, X_2, \ldots, X_n\) be iid of a \(N(\mu, \sigma^2)\). Given that \(\sigma^2\) is known, find the Bayesian estimator for \(\mu\) with a conjugate prior \(\mu \sim N(\phi, \gamma)\).

3. Let \(\tau = \frac{1}{\sigma^2}\), and \(X_1, X_2, \ldots, X_n\) be iid of a \(N(\mu, \sigma^2 = 1/\tau)\). Find the conjugate prior for \(\tau\).

4. Using the conjugate prior for Problem 3, find the Bayesian estimator for \(\tau\), assume that \(\mu\) is known.

5. Let \(X_1, X_2, \ldots, X_n\) be iid of an \(Exponential(\lambda)\). Find the method of moment estimator for \(\lambda\).