Assume that $X$ is a random variable from a distribution with density function $f$. Find the most powerful test of size-$\alpha$ to test $$H_0:X\sim G(1,1)$$ $$H_1: X\sim N(1,1)$$
I know that I need to use Neyman Pearson Lemma usually to solve this kind of question. But normally I am given the parameter (like $\theta$) for this type of questions. But now I am not really sure how to proceed.
I would appreciate any help or hints.
You can think of the situation like this:
$$X\sim \theta \,\Gamma(1,1)+(1-\theta)N(1,1)\,,\quad\theta\in\{0,1\}$$
You are to test $H_0:\theta=1$ against $H_1:\theta=0$, i.e. a simple null versus a simple alternative.
But to actually solve the problem, there is no need to introduce any parameter $\theta$.
Hint:
Follow the standard procedure. Let $f$ be the density of $X$. Find the likelihood ratio $f_{H_1}/f_{H_0}$, where $f_{H_j}$ is the pdf of $X$ under $H_j$, $j=0,1$. By Neyman-Pearson lemma, a most powerful test rejects $H_0$ whenever the ratio $f_{H_1}/f_{H_0}$ is large. Simplify the rejection region from there and hence find the exact test.