Suppose that $X_1,X_2,\ldots , X_n$ for a random sample from a Poisson distribution with parameter $\lambda$. Propose an unbiased estimator for $\theta = e^{-\lambda}$
My attempt was to substitute this estimator in the Poisson distribution $f(x|\lambda)$ = $\frac{\lambda^{x} e^{-\lambda}}{x!}$ obtaining $\frac{\log(\theta^{x}) \theta}{x!}$. I don't know if this is the right approach and I wouldn't know how to find an unbiased estimator afterwards. I thought about calculating the MLE of this new distribution, but I don't know if it was correct.
Any help/hint would be appreciated.