Most powerful size $\alpha$ test

Question

Someone can help me to check this answer?

How to find the Most Powerful Test size $\alpha$ and Power of Test, Since I have $H_0 : X \thicksim f_{\theta 0}= (1/\sqrt(2\pi) \exp^{(-x^2/2)}$ and $H_1 : X \thicksim f_{\theta 1}= 1/2 \exp^{-|x|}$ with $ -\infty <x< \infty$ based on sample size 1.

My answer is:

to find the Most Powerful Test size we just calculate $$f_{\theta 1}/f_{\theta 0}= \sqrt{\pi/2}\exp^{-|x|+x^2/2}$$

then the power of test is given by ( I am not sure about the result of my integration)... $$E_1 \varphi(x)=1-\int_{-k}^k(1/2)\exp^{-|x|} dx = 1.$$

Thanks.

Answer 1

I don't quite understand the problem, but regardless of the context, the integral

$$\int_{-k}^{k} \frac{1}{2}\mathrm{Exp} (-|x|) dx = \frac{1}{2}(2-2e^{k})$$ for real $k$.