# Question

$H_0:\ \mbox{X has PDF}\ f_{X|\theta}(x|\theta_0)= \begin{cases} 2x & \mbox{for }0 \le x \le 1 \\ 0 & \mbox{else} \end{cases}$

$H_1:\ \mbox{X has PDF}\ f_{X|\theta}(x|\theta_1)= \begin{cases} 1 & \mbox{for }0 \le x \le 1 \\ 0 & \mbox{else} \end{cases}$

Find the probability of type I errors and type II errors using the ML rule.

To find the ML Rule we say pick $H_1\!$ if $f_{X|\theta}(x|\theta_1)>f_{X|\theta}(x|\theta_0)\!$

Or in otherwords pick $H_1\!$ if $1>2x\!$ Thus,

$\mbox{ML Rule: } \begin{cases} \mbox{say }H_1 &\mbox{if }x<1/2 \\ \mbox{say }H_0 &\mbox{else} \end{cases}$

Type I Error: A Type I error is the $Pr[\mbox{say } H_1|H_0]\!$ this is equivalent to saying $Pr[x<1/2|\theta=\theta_0]\!$ we calculate this using integration

$Pr[x<1/2|\theta=\theta_0] = \int_{0}^{1/2}2x\, dx=1/4\!$

Type II Error:A Type II error is the $Pr[\mbox{say } H_0|H_1]\!$ this is equivalent to saying $Pr[x\ge1/2|\theta=\theta_1]$ we calculate this using integration

$Pr[x\ge1/2|\theta=\theta_1] = \int_{1/2}^{1}1\, dx=1/2\!$

## Alumni Liaison

Correspondence Chess Grandmaster and Purdue Alumni

Prof. Dan Fleetwood