State and prove the Neyman-Pearson lemma.

Suppose that $X_{1}, \ldots, X_{n}$ are i.i.d. $\exp (\lambda)$ random variables where $\lambda$ is an unknown parameter. We wish to test the hypothesis $H_{0}: \lambda=\lambda_{0}$ against the hypothesis $H_{1}: \lambda=\lambda_{1}$ where $\lambda_{1}<\lambda_{0}$.

(a) Find the critical region of the likelihood ratio test of size $\alpha$ in terms of the sample mean $\bar{X}$.

(b) Define the power function of a hypothesis test and identify the power function in the setting described above in terms of the $\Gamma(n, \lambda)$ probability distribution function. [You may use without proof that $X_{1}+\cdots+X_{n}$ is distributed as a $\Gamma(n, \lambda)$ random variable.]

(c) Define what it means for a hypothesis test to be uniformly most powerful. Determine whether the likelihood ratio test considered above is uniformly most powerful for testing $H_{0}: \lambda=\lambda_{0}$ against $\widetilde{H}_{1}: \lambda<\lambda_{0}$.