Suppose we have a random sample $X_1$, $X_2$ from the Beta($\theta$, $1$) distribution and we want to test $H_{\theta} :\theta \leq 1$ against $H_1:\theta > 1$. The following test issued: “Reject $H_0$ if and only if $3X_1 \leq 4X_2$.” How to show that the power function of the test is given by $\beta(\theta) = 1-\frac12\left(\frac34\right)^\theta$
My try :
This is not related to question, but I know how to solve if it is only one observation: let's say $X_1$ ~ Beta($\theta$, $1$) and the condition is $X_1 > \frac{1}{2}$ for same $H_0$ and $H_1$. To get the power function we have to solve for :
$\beta(\theta) = P_\theta(X > 1/2) = \int_{1/2}^1 \frac{\Gamma(\theta + 1)}{\Gamma(\theta)\Gamma(1)}x^{\theta-1}(1-x)^{1-1}\mathrm dx$
I do not know how to go about the original problem I mentioned in the question.