1. ## Hypothesis Testing

Let $X$ have a pdf of the form $f(x; {\theta}) = {\theta}x^{\theta-1}$ where ${\theta}\in\{\theta: \theta=1, 2\}$.

To test the simple hypothesis $H_0:\theta=1$ against the alternative simple hypothesis $H_1:\theta=2$, use a random sample $X_1, X_2$ of size $n=2$ and define the critical region to be $C = \{(x_1, x_2): \frac{3}{4} \leq x_1x_2 \}$.

Find the power function of the test.

I hate to ask for help when I feel like I don't have much of anything to show for my own efforts, but I feel stuck.

I know I need $\gamma(\theta) = P_{\theta}[\frac{3}{4} \leq x_1x_2] = 1 - P_{\theta}[x_1x_2 \leq \frac{3}{4}]$

I don't know how to set the limits of integration on the joint pdf to get the answer given in the back of the book:
$1-(\frac{3}{4})^\theta+\theta(\frac{3}{4})^{\theta}l og({\frac{3}{4})}
$