I've got a one major problem using the Neyman-Pearson lemma.
We're testing hypotheses $H_0: \theta \le \theta_0$ vs. $H_1: \theta > \theta_0$. Our $$f(x,\theta) = \frac{1}{\theta}e^{-\frac{x}{\theta}}$$
We use the Neyman-Pearson lemma and get the following expression for the critical value of the test: $$\phi(X) = \mathbf{1}_{\{f(x,\theta_1) > k f(x,\theta_0)\}}(x) + \gamma \mathbf{1}_{\{f(x,\theta_1) = k f(x,\theta_0)\}}(x)$$
Now I can get some expression in terms of $\frac{1}{n} \sum_{i=1}^n X_i > g(k)$ for the test, but I need a value for $k$ from the expression for $\phi(X)$.
How would I compute $$\mathbb{E}[\phi(X) |H_0] =\alpha$$ so that I can solve for $k$?