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I was working on the following problem:

Consider two probability density functions on $[0,1]: f_0(x) = 1$, and $f_1(x) = 2x$. Among all tests of the null hypothesis $H_0: X \sim f_0(x)$ versus the alternative $X \sim f_1(x)$, with significance level $\alpha = 0.1$, how large can the power possibly be?

I think we need to begin by looking at an arbitrary test with significance level $\alpha = 0.1$ but I am having a hard time doing this. I am not even sure this is the direction we want to head in so I was hoping to get some hints regarding this problem.

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Let $\{\Gamma_0, \Gamma_1\}$ be a partition of $[0,1]$ into (measurable) sets and the decision rule be that hypothesis $H_i$ is true if the observation $X$ belongs to $\Gamma_i$. Then, the false alarm probability is $$\alpha = \int_{\Gamma_1} f_0(x)\,\mathrm dx = 0.1$$ while the power of the test is $$\beta = \int_{\Gamma_1} f_1(x)\,\mathrm dx.$$ Now,since $f_0(x)$ is the uniform density, we know that the total "length" of the set $\Gamma_1$ is $0.1$ and so the question is

What should we choose $\Gamma_1$ to be so as to maximize $\displaystyle \int_{\Gamma_1} f_1(x)\,\mathrm dx$?

Sketching the shape of the density $f_1(x)$ might reveal the answer to you in a flash, and then you can write down a more formal way of getting to it if you like.

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