Sufficiency and UMVUE for Poisson distribution

Question

I need to show that $\hat\lambda = \bar X$ is a sufficient estimator for a Poisson distribution iid $X_1...X_n$, show that $\hat\lambda$ is the UMVUE for $\lambda$ and that $\hat\lambda$ is a consistent estimator.

I don't even know how to tackle the sufficiency part. I've looked over the definition and I don't understand at all.

For the UMVUE I already have that var($\hat\lambda$)=CRLB, but I can figure out how to evaluate E($\hat\lambda$) at all.

I have a terrible head cold, so these may be stupid questions but any help is appreciated!!

Answer 1

I don't even know how to tackle the sufficiency part.

http://en.wikipedia.org/wiki/Sufficient_statistic#Fisher.E2.80.93Neyman_factorization_theorem

recognize that

$$\prod \lambda^{x_i} = \lambda^{\sum x_i}$$

After which it's trivial

I can figure out how to evaluate $\operatorname{E}(\hat{\lambda})$ at all

Write $\bar{X}$ as a constant times a sum. What's the distribution of that sum?

Can you find the expectation of that?