Uniformly Most Powerful Tests.

[Artusartos]

Homework Statement

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.

Homework Equations

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...

 

[jashua]

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