Cramér–Rao Lower Bound for a Function of a Parameter

I'm trying to compute the Cramér–Rao Lower Bound for a function of the parameter $\displaystyle \lambda$ of a Poisson distribution.

The function is $\displaystyle g (\lambda) = e^{-\lambda} $.

Let $\displaystyle \hat{g} $ be an estimate of $\displaystyle g(\lambda) $

I'm using the formula:

$\displaystyle Var (\hat{g}) \ge \dfrac{(g'(\lambda))^2}{nI(\lambda)} $

Now from estimating $\displaystyle \lambda $ previously, I know that $\displaystyle nI(\lambda) = \dfrac {n}{\lambda} $

And

$\displaystyle g'(\lambda) = -e^{-\lambda} $

$\displaystyle (g'(\lambda))^2 = e^{-2\lambda} $

So

$\displaystyle Var (\hat{g}) \ge \dfrac{e^{-2\lambda}}{\dfrac {n}{\lambda}} $

$\displaystyle Var (\hat{g}) \ge \dfrac{\lambda e^{-2\lambda}}{n} $

Is this correct because I tried it another way and got?

$\displaystyle Var (\hat{g}) \ge \dfrac{e^{-\lambda}(1-e^{-\lambda})}{n} $