a sample of n i.i.d poisson variates has distribution

we know the maximum likelihood estimate is minimum variance, we need to see if it's unbiased.

$\large p_{\vec{K}}(\vec{k})=\displaystyle{e^{-\lambda} \prod_{i=1}^n}\dfrac {\lambda^{k_i}}{k_i!}$

$\large \ln(p_{\vec{K}}(\vec{k})=\displaystyle{\sum_{i=1}^ n}\left(k_i \ln(\lambda)-\ln(k_i!)-\lambda\right)$

$\large \dfrac \partial {\partial \lambda}\ln(p_{\vec{K}}(\vec{k}))=\dfrac 1 \lambda \displaystyle{\sum_{i=1}^n}\left(k_i - 1\right)$

setting this to zero we get

$\dfrac 1 \lambda \displaystyle{\sum_{i=1}^n}k_i = n$

$\dfrac 1 n \displaystyle{\sum_{i=1}^n}k_i = \lambda$

as $E[k_i]=\lambda$ we see that it is unbiased and thus the mean is our unbiased minimum variance estimate.