Let $X_{1}, \ldots, X_{n}$ be random variables with joint density function $f\left(x_{1}, \ldots, x_{n} ; \theta\right)$, where $\theta$ is an unknown parameter. The null hypothesis $H_{0}: \theta=\theta_{0}$ is to be tested against the alternative hypothesis $H_{1}: \theta=\theta_{1}$.

(i) Define the following terms: critical region, Type I error, Type II error, size, power.

(ii) State and prove the Neyman-Pearson lemma.