# Uniformly Most Powerful Tests.

1. Mar 26, 2013

### Artusartos

1. The problem statement, all variables and given/known data

Let X have the pdf $f(x, \theta) = \theta^x(1-\theta)^{1-x}$, x = 0, 1, zero elsewhere. We test $H_0 = \theta = 1/2$ against $H_1 : \theta < 1/2$ by taking a random sample $X_1, X_2, ... , X_5$ of size n=5 and rejecting $H_0$ if $Y = \sum^n_1 X_i$ is observed to be less than or equal to a constant c. Show that this is a uniformly most powerful test.

2. Relevant equations

3. The attempt at a solution

$L(\theta; x_1, ... ,x_n) = \theta^{x_1}(1 - \theta)^{1-x_1}...\theta^{x_n}(1-\theta)^{1-x_n}$

$\frac{L(\theta'; x_1, ... ,x_n)}{L(\theta''; x_1, ... ,x_n)} \leq k$

$\frac{\theta'^{x_1}(1 - \theta')^{1-x_1}...\theta'^{x_n}(1-\theta')^{1-x_n}}{\theta''^{x_1}(1 - \theta'')^{1-x_1}...\theta''^{x_n}(1-\theta'')^{1-x_n}}$

$(\frac{\theta'}{\theta''})^{x_1+...+x_n}(\frac{1-\theta'}{1-\theta''})^{n-\sum^n_1 x_n} \leq k$

Now I was trying to transform the left side of the equality into a binomial distribution, but I was kind of stuck...

2. Mar 26, 2013

### jashua

Why do you need to obtain a binomial distribution? The question does not require this.