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Artusartos
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Homework Statement
Suppose a sample of size 10 is drawn from a distribution with probability density function ##f(x, \theta) = 2x^{\theta}(1-x)^{1-\theta}## if ##0<x<1## and ##0## otherwise, where ##\theta \in \{0,1\}##. Describe a best critical region of size ##\alpha## for testing ##H_0 : \theta = 0## against the alternative hypothesis ##H_1 : \theta =1##.
Homework Equations
The Attempt at a Solution
We need ##\frac{L(0)}{L(1)} \leq k## for some ##k < 1##
We find that ##\frac{L(0)}{L(1)} = \frac{2(1-x_1)2(1-x_2)...2(1-x_{10})}{2x_12x_2...2x_{10}} = \frac{(1-x_1)(1-x_2)...(1-x_{10})}{x_1x_2...x_{10}}##. Now I want to simplify ##\frac{(1-x_1)(1-x_2)...(1-x_{10})}{x_1x_2...x_{10}} \leq k## to see if I can turn this into a distribution that looks more familiar. But for some reason I wasn't able to do anything, so I was wondering if somebody would give me a hint so I can continue...
Thanks in advance
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