$1 . \mathrm{I} . 3 \mathrm{H} \quad$

State the factorization criterion for sufficient statistics and give its proof in the discrete case.

Let $X_{1}, \ldots, X_{n}$ form a random sample from a Poisson distribution for which the value of the mean $\theta$ is unknown. Find a one-dimensional sufficient statistic for $\theta$.

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