Sufficient Estimator for a Geometric Distribution

1. Jan 12, 2010

cse63146

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

Let X1,..,Xn be a random sample of size n from a geometric distribution with pmf $$P(x; \theta) = (1-\theta)^x\theta$$. Show that $$Y = \prod X_i$$ is a sufficient estimator of theta.

2. Relevant equations

3. The attempt at a solution

So $$\prod P(x_i, \theta) = (1-\theta)^{\Sigma x_i} \theta^n$$

I don't believe that the factorization theroem can be applied here. Is there some trick to this that I'm not seeing?

Thank you in advance.