Give, $Y_1, Y_2, ... Y_n$ are IID random variables, each having a Poisson Distribution with parameter $\lambda$. Is there any unbiased estimator of either (i) $e^{-\lambda}$ or (ii) $e^{-2\lambda}$ whose variance can achieve the Cramer- Rao bound corresponding to them? For $e^{-\lambda}$ , I found the CRLB to be $\dfrac{\lambda}{n}e^{-2\lambda}$ and for $e^{-2\lambda}$ it was $\dfrac{4\lambda}{n}e^{-4\lambda}$.
Could anyone lead me further with this?