# Uniformally most powerful test

let x be a random variable with density function

f(x)= $\frac{2\theta x + 1}{\theta + 1}$ if $0\le x \le 1$,$\theta$ > -1 and 0 otherwise

consider the problem of testing $H0 :\theta \le 1$  $against$ $H1:$ $\theta >1$

let $\phi$ be the test function given by $\phi(x)$ = 1 if x $\ge$ $\frac{\sqrt{9-8\alpha}\ -1}{2}$ and 0 otherwise then we have to prove that $\phi$ is UMP size $\alpha$ test

what i tried :

first i checked the monotone likelihood ratio and infered that UMP will exist ,.,., till then it was OK

now for $\phi$ to be UMP size $\alpha$ test we must have E ( $\phi$(x) ) under H0 = $\alpha$

[rest is shown in the image] [ ]1

after this i am stuck , i have rechecked my integration it seems to be correct any suggestions / corrections please

• Could you please learn mathjax and replace the scanned formulas with typed formulas, that is, with $\LaTeX$? – kjetil b halvorsen Jan 7 '17 at 16:22
• Although I agree with @kjetilbhalvorsen, for the sake of our vision impaired users, your handwriting is very beautiful! – Matthew Drury Jan 7 '17 at 16:50
• Since this looks like homework add the self study tag. – Michael Chernick Jan 7 '17 at 19:45
• You haven't yet used the hypothesis, which suggests plugging in $\theta=1$. The integration isn't quite correct: double-check the signs and the constant term in the numerator. – whuber Jan 7 '17 at 23:18
• 2. @ Matthew Drury , thanks very much , thats why i posted the image coz i can write clearly , but i apologize for not being good at english grammar . – ANUJ NAIN Jan 8 '17 at 2:34