State and prove the Neyman-Pearson lemma.

Suppose that $X$ is a random variable drawn from the probability density function

$$f(x \mid \theta)=\frac{1}{2}|x|^{\theta-1} e^{-|x|} / \Gamma(\theta), \quad-\infty<x<\infty$$

where $\Gamma(\theta)=\int_{0}^{\infty} y^{\theta-1} e^{-y} d y$ and $\theta \geqslant 1$ is unknown. Find the most powerful test of size $\alpha$, $0<\alpha<1$, of the hypothesis $H_{0}: \theta=1$ against the alternative $H_{1}: \theta=2$. Express the power of the test as a function of $\alpha$.

Is your test uniformly most powerful for testing $H_{0}: \theta=1$ against $H_{1}: \theta>1 ?$ Explain your answer carefully.