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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.

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Paul Ash
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  • You may want to double check your work, because $X_1 + X_2$ must be non-negative, but $-\frac{2}{3}\ln 2 < 0$, so $P\left(X_1 + X_2 \le -\frac{2}{3}\ln 2\right) =0$. – Minus One-Twelfth Mar 10 '19 at 04:20
  • Yes you were correct, my algebra was off. What I was supposed to end up with was $P(X_1 + X_2 \le 2ln(2))$. From here, I had to figure out that for $\theta = 2$, both $X_1$ and $X_2$ are $\Gamma(1,2)$, and when you add two independent gamma random variables, you get $\Gamma(\alpha_1 + \alpha_2, \beta)$, so here, $X_1 + X_2$ follows a $\Gamma(2,2)$ – Paul Ash Mar 10 '19 at 19:27

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