I have 2 independent variables $X$ and $Y$ which are both normal with means $\theta x$ and $\theta y$ respectively and variance=1. I need to find the most powerful test of size $\alpha=0.1$ for the following hypotheses:
Ho: $(\theta x, \theta y)$=$(0,0)$
H1: $(\theta x, \theta y)$=$(1,0)$
My approach is as follows. First, I used the given that $X$ and $Y$ are independent and normal to get their joint pdf as just the product of their individual pdfs:
$f[(x,y)|(\theta x, \theta y)]=(1/2\pi)^2\exp[-[(x-\theta x)^2+(y-\theta y)^2)/2]]$
Then, using the Neyman-Pearson Lemma, I took the ratio of the joint pdf under the alternative hypothesis by the joint pdf under the null which ended with the expression: $\exp(x-1/2)$
Does this mean that using the NP lemma and letting $\exp(x-1/2)>k$, the test that rejects Ho if $x >lnk + 1/2$ is the most powerful test?
Also, in order to achieve size $\alpha=0.1$, I computed for $k$ such that $P(x > lnk +1/2)$ which turns out to be 2.18486. Does this mean that the most powerful test of size $\alpha$=0.10 is the test where Ho is rejected when $x > 1.2815$?
Finally, I noticed that the ratio using NP lemma simplified to just $x$ ($y$ was cancelled out). Is this because $\theta y$ is the same in both the null and the alternative? If so, would I get the same answer if I just removed $y$ from the start and worked with just one variable?