Homework 3

Homework 3 is due 3/29/2026 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

   \[ f(x) = \left\{\begin{array}{cc} \frac{1}{\theta}2xe^{-x^2/\theta} & 0<x;0< \theta \\ 0 & \mathrm{elsewhere} \end{array} \right. \]

   Find the MLE for \(\theta\).

2. Is the MLE estimator from Problem 2 unbiased?

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

   \[ f(x) = \left\{\begin{array}{cc} (\theta + 1)x^\theta& 0\le x\le 1;\theta>-1 \\ 0 & \mathrm{elsewhere} \end{array} \right. \]

   Find the MLE for \(\theta\).

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

   \[ f(x) = \left\{\begin{array}{cc} \frac{1}{\Gamma(\alpha)\theta^\alpha}x^{\alpha-1}e^{-x/\theta} & 0<x;0< \theta \\ 0 & \mathrm{elsewhere} \end{array} \right. \]

   where \(\alpha>0\) is known. Find the MLE for \(\theta\).