Homework 6

Homework 6 is due 4/13/2023 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

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

   Find the MLE for $\theta$.

2. 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 $\theta$.

3. Let $X_1, \ldots, X_n\overset{iid}{\sim}Geo(p)$, what is the MLE of $p$.