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On my last exam, this was one of the question I got wrong. However this was the only question I didn't manage to figure out even after the exam. Can anyone help me? With part c) especially?

In order to test the null hypothesis that X has a uniform distribution on the interval $(0,1)$ against the alternative that X has a triangular distribution $[f(x) = 2x$ for $0 < x < 1]$ a random sample of size n is chosen. You want to find the most powerful test at significance level $\alpha = 0.10$

a) If $n = 1$, for what values of X do you reject the null hypothesis?

b) What is the power of the test in a)?

c) If n = 10, for what observations do you reject the null hypothesis?

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Since this is a simple vs simple hypothesis test, use NP lemma with $$H_0: f(x)=\,\mathbb{1}_{(0,1)}(x)\quad \text{ against }\quad H_1: f(x)=2x \,\mathbb{1}_{(0,1)}(x).$$ For part c, the likelihood ratio is $$ L = \frac{f_1(x_1,\ldots,x_{10})}{f_0(x_1,\ldots,x_{10})} = 2^{10} x_1\cdots x_{10}. $$ So the rejection region is the set of $x$ for which \begin{align*} L > c &\iff 2^{10} x_1\cdots x_{10} > c \\ &\iff \sum_{i=1}^{10} (-\log x_i) < c'. \end{align*} To find $c'$, notice that $$X_i\sim Unif(0,1) \implies -\log X_i \sim Exp(1)= Gamma(1,1)$$ and $$Y_1,\ldots,Y_n \sim Gamma(1,1) \,\,iid \implies \sum_{i=1}^n Y_i \sim Gamma(n,1).$$

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