The RV Y have PDF $$f(y)=\theta y^{\theta - 1}$$ for $0 \le y \le 1, \theta > 0$. Given only one obseravtion of Y, find the uniformly most powerful test with sign level $\alpha$ of $H_0: \theta = 1, H_1: \theta > 1$.
Shoudl I start by finding the distribution of Y given its PDF, or straight at it using the definitions?
Attempt: We know $L(\theta) = \prod f(y_i)$. Set $$L(\theta_0)/L(\theta_1)=k$$ then take logs, and solve for $y_i$ or $\sum y_i$. But I cannot do the algebra.
Note: $Y\sim Beta(\theta,1)$
One observation case: By NP-lemma Rejection region is the following $$R=\{y:y>k_{\alpha}\}$$ But what is $k_{\alpha}$?. This is determined by the size of the test. i.e. epuating $P_{H_0}(R)=\alpha$. Or in other words $P_{\theta=1}(Y>k_{\alpha})=\alpha$ where $Y\sim Beta(1,1)=U(0,1)$. So life is really a bed of roses in this case because $P_{\theta=1}(Y>k_{\alpha})=1-k_{\alpha}$ and equating that to $\alpha$ we get $k_{\alpha}=1-\alpha$. Hence reject $H_0$ if $y>1-\alpha$.
The more interesting problem will be if we've more that 1 observations i.e. $Y_i \overset{iid}{\sim} Beta(\theta,1)$. If you need any help regarding that let us know.