Let $X_1$, $X_2$ be a random sample of size $n=2$ from the distribution having pdf $$f(x;\theta)=\left( \dfrac 1{\theta} \right)e^{-\frac x{\theta}}, 0 \lt x \lt \infty$$
We reject $H_0: \theta=1$ is the observed values of $X_1$, $X_2$, say $x_1$, $x_2$, are such that
$$\dfrac {f(x_1;2)f(x_2;2)}{f(x_1;1)f(x_2;1)} \le \dfrac 12$$ Here, $\Omega=\{ \theta: \theta=1,2\}$. Find the significance level of the test and the power of the test when $H_0$ is false.
Okay, so I basically did the annoying algebra and ended up with trying to find the significance level :|
$\alpha=P\left[ X_1+X_2 \le -\frac 23 \ln (2)\right]$. I forgot how to interpret this probability. I know that $X_1$ and $X_2$ are independent, but I'm just stuck here.