constructing the UMP $\alpha$ = .10 size test of $H_{0}$ :$ \lambda \leq 1$ versus $H_{1}:\lambda > 1$.

Question

Given that a sample size n =10, from a poisson($\lambda$), construct the UMP $\alpha$ = .10 size test of $H_{0}$ :$ \lambda \leq 1$ versus $\lambda > 1$

For this question I have found that we have a MLR property in T =$\sum_{i=1}^{10}X_{i}$. In addition, using Blackwell-Girshick theorem, I can find the UMP in the form

$ \phi^{*} = \begin{cases} 1 \text{ if } T>k\\ \gamma \text{ if } T =k\\ 0 \text{ if } T < k \end{cases} $

My question is how do I find the value of k. I know that I need to find the distribution of T first, but I am stuck on how to write the working out.

Answer 1

Maybe you mean Karlin-Rubin Theorem? But you find $k$ so that $E[ \phi^* ; H_0 \text{ is true}] = \alpha$.