ii. Let $X_1, X_2, \ldots, X_n$ be iid Poisson($\lambda$). Find a Uniformly Most Powerful test of

$$H_0: \lambda \le \lambda_0 \text{ v.s. } H_1: \lambda > \lambda_0.$$

iii. Let $X_{ij} \sim f(x_{ij}|\theta_i) = \frac{1}{\theta_i} e^{\frac{x_{ij}}{\theta_i}}, i = 1, 2, j = 1, 2, \ldots, n$ be 2 independent random samples from two exponential populations. Find a likelihood ratio test of

$$H_0: \theta_1 = \theta_2 \text{ v.s. } H_1: \theta_1 \neq \theta_2$$