most powerful test using Neyman-Pearson lemma with two parameters
From Wackerly, Mathematical Statistics with Applications, 7th Ed. (p548):
According to the likelihood function,
denote a random sample from a population having a Poisson distribution with mean
denote an independent random sample from a population having a Poisson distribution with mean
. Derive the most powerful test for testing
So by Neyman-Pearson, our rejection region for the most powerful test is given by
Substituting , we have the rejection region
where we define by
and is the desired probability of a type I error, that is, the "level" of the test.
Assuming I have not made any errors in the above analysis, then I am curious, can I do anything more towards defining ? In particular, is there any way to define as a function of ?