The discrete random variable X has the following probability distributions under $H_0$ and $H_1$
$$\begin{array}{r|rrrrrrrrrr} x&1&2&3&4&5&6&7&8&9&10\\ \hline p(x)\;\mbox{under}\;H_0&0.63&0&0.09&0.08&0.07&0.01&0.02&0.03&0.05&0.02\\ p(x)\;\mbox{under}\;H_1&0&0.42&0.12&0.05&0.09&0.01&0.04&0.08&0.12&0.07 \end{array}$$
A single observation is to be made on X. Identify the best (most powerful) test of fixed level (exactly) α = 0.10 for discriminating between $H_0$ and $H_1$. [Hint: Consider all possible tests which have α = 0.10 and compare their power. Remember, a test is specified by specifying its critical region.]
Edit: I think I can calculate the power if I have the critical region for both hypotheses. But I don't know how to find the critical region when the probability distribution is discrete.
Can anyone help?
Thanks