Let X be a random variable having a probability density function $f \in (f_o,f_1)$, where
$f_0(x) = \biggl(1 \ \ \ \ if \ \ 0\leq x \leq 1\,, \ 0 \ otherwise\biggr )$
and
$f_1(x) = \biggl(\dfrac{1}{2} \ \ \ \ if \ \ 0\leq x \leq 2\,, \ 0 \ otherwise\biggr )$
For testing the null hypothesis $H_0:f=f_0$ against $H_1:f=f_1$ based on a single observation on $X$, the power of the most powerful test of size $\alpha = 0.05$ equals ?
My inpuit : I applied NP lemma. Since we have a single observation. Test is to reject $H_0$ if $Pr$$\left\{\dfrac{f_1}{f_0}\geq K \right\}$ = $.05$
I am having problem in finding the critical region. I divided these densities and i got $\dfrac{f_1}{f_0} = \dfrac{1}{2} \ \ \ \ \ \ \ 0\leq x \leq 1\ $
$\dfrac{1}{2} \geq K$
What do i depict from it ?
Let $\Lambda(x) = \frac{f_0(x)}{f_1(x)}$ and reject if $\Lambda(x) < \tau$. Note that because the likelihood ratio has the Monotone Likelihood Ratio Property we can instead reject if $x>\tau'$.
Letting $\phi(x) = I_{x>.95}$ be our test. We can see that $\alpha = E_0[\phi(x)] = P_0[X>.95] = .05$.
To get the power of the test: $E_{1}[\phi(x)] = P_1[X>.95] = \frac{1}{2}[2-.95] = \frac{21}{40}$.
Because the test was constructed using the Neyman-Pearson Lemma, and because there is only one parameter in the alternative set, it is a UMP test.