State and prove the Neyman-Pearson lemma.

A sample of two independent observations, $\left(x_{1}, x_{2}\right)$, is taken from a distribution with density $f(x ; \theta)=\theta x^{\theta-1}, 0 \leqslant x \leqslant 1$. It is desired to test $H_{0}: \theta=1$ against $H_{1}: \theta=2$. Show that the best test of size $\alpha$ can be expressed using the number $c$ such that

$$1-c+c \log c=\alpha .$$

Is this the uniformly most powerful test of size $\alpha$ for testing $H_{0}$ against $H_{1}: \theta>1 ?$

Suppose that the prior distribution of $\theta$ is $P(\theta=1)=4 \gamma /(1+4 \gamma), P(\theta=2)=$ $1 /(1+4 \gamma)$, where $1>\gamma>0$. Find the test of $H_{0}$ against $H_{1}$ that minimizes the probability of error.

Let $w(\theta)$ denote the power function of this test at $\theta(\geqslant 1)$. Show that

$$w(\theta)=1-\gamma^{\theta}+\gamma^{\theta} \log \gamma^{\theta}$$