What is the critical region $C$ of a test of the null hypothesis $H_{0}: \theta \in \Theta_{0}$ against the alternative $H_{1}: \theta \in \Theta_{1}$ ? What is the size of a test with critical region $C ?$ What is the power function of a test with critical region $C$ ?

State and prove the Neyman-Pearson Lemma.

If $X_{1}, \ldots, X_{n}$ are independent with $\operatorname{common} \operatorname{Exp}(\lambda)$ distribution, and $0<\lambda_{0}<\lambda_{1}$, find the form of the most powerful size- $\alpha$ test of $H_{0}: \lambda=\lambda_{0}$ against $H_{1}: \lambda=\lambda_{1}$. Find the power function as explicitly as you can, and prove that it is increasing in $\lambda$. Deduce that the test you have constructed is a size- $\alpha$ test of $H_{0}: \lambda \leqslant \lambda_{0}$ against $H_{1}: \lambda=\lambda_{1}$.