# Poisson distribution, likelihood ratios

1. Dec 10, 2006

### Auron87

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

An independent, identically distributed sample, x = (x1, ... , xn) of size n, is drawn from a Poisson distribution, parameter A. We want to test the null hypothesis H0 : A = A1 against the alternative hypothesis H1 : A = A2 where A1 < A2.

Write down the likelihood ratio for the data, and show that all likelihood ratio tests of H0 against H1 are of the form: - Reject H0 if the sum from one to n of xi > c.

2. Relevant equations

LR(x) = f1(x)/f2(x)

3. The attempt at a solution

f0(xi) = (e^(-A1)*A1^xi) / xi!

f1(xi) = (e^(-A2)*A2^xi) / xi!

Then I'm not sure if this is right but I thought each f(x) is equal to the product of its f(xi)

so I have f0(x) = (e^(-A1)*A1^(sum of xi's from 1 to n)) / x1! ... xn!

Sorry don't know if I can put it sigma signs but don't know how and then obviously f1(x) is the same but with A2 instead.

So that gave me after some simplifying

LR(x) = e^(A1 - A2)*(A2/A1)^(sum of xi's)

Now I'm not sure if this is quite right or if I can simplify it further? Any help is much appreciated, thankyou.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution