Let $X_1, X_2, ..., X_n$ be an iid random sample from a Poisson$(\lambda)$ distribution:
a) find the UMVUE of $\theta$ = $\lambda^k$ for $k > 0$ a known integer
b) find the UMVUE of $\tau = e^{-\lambda} = \mathbb{P}(X = 0)$.
Edit: I have found that $T = \sum X_i$ is a complete sufficient statistic. It has a Poisson$(n\lambda)$ distribution. Now to find the UMVUE, we need a function $g(t)$ such that $E[g(T)] = \theta$, then $g(t)$ will be UMVUE.
For, that $E[g(T)] = \theta$ implies
$$e^{-n\lambda}\sum_{t=0}^\infty g(t) \frac{(n\lambda)^t}{t!} = \theta =\lambda^k.$$
Now I don't know how to proceed here. If it were the continuous case, I could use the differential regularity condition. The same thing in b) too.