Sufficient Estimator for a Geometric Distribution

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Homework Statement



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

Homework Equations





The Attempt at a Solution



So [tex]\prod P(x_i, \theta) = (1-\theta)^{\Sigma x_i} \theta^n[/tex]

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.
 

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