# Homework 6

**Homework 6 is due 4/13/2023 at 11:59 PM. Submit your homework on Canvas as one PDF document.**

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.

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.

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