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!!