I have some problems trying to solve the following problem:
Let $f_{0}$ and $f_{1}$ be pdfs for discrete integer-valued random variables, where
$f_{0}(n) = \frac{1}{6} \mathbb{1} [n ∈ [0,1,2,3,4,5]]$
and $f_{1}$ is the pdf for a Poisson random variable with mean $1$. Let $ \Theta = \{ 0, 1 \}$. Consider the null hypothesis $H_{0} : \theta = 0$. Let $X$ be a random variable with distribution $f_{\theta}$, where $\theta \in \Theta$ is unknown. Based on observation, $X$, find the the best test at levels $\alpha = \frac{1}{3}$ and $\alpha = \frac{3}{9}$. Find the power of your tests.
I'm still not sure what procedure to follow so any help will be much appreciated.