Let $X = (X_1,\ldots,X_n)$ be a random sample, where $X_1 \sim \mathrm{Bern}(p)$. Let $\lambda = e^p$.
Question: By law of large numbers, $T=e^{(\bar{X})}$ is a consistent estimator for $\lambda$, where $\bar{X}$ is the sample mean. Using delta method, estimate (for large values of $n$), the variance of $T$. Does there exist an unbiased estimator for $\lambda$? That is, does there exist an estimator $Y$ such that $\mathbf{E}_{p} Y=e^{p}$ for all $p\in (0,1)$?
I'm not sure about using just the variance of the $\mathrm{Bernoulli}(p)$ as just $p(1-p)$, and that's it or if there is something that I am missing to this problem.