most powerful test using Neyman-Pearson lemma with two parameters

From Wackerly, *Mathematical Statistics with Applications*, 7th Ed. (p548):

Quote:

Let

denote a random sample from a population having a Poisson distribution with mean

. Let

denote an independent random sample from a population having a Poisson distribution with mean

. Derive the most powerful test for testing

versus

.

According to the likelihood function,

.

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 ?

Thanks!