Let $X \sim Poiss(\lambda)$.
As, $\displaystyle \sum_{i=1}^{N} X_i $ is sufficient statistic for both mean (and variance) of $Y$, so we can define the unbiased estimate for mean as , $ s=\frac{1}{N} \sum_{i=1}^{N} X_i$, where all samples are i.i.d.
I am wondering how to find the estimator if parameter of interest is second moment i-e $E[X^2]=\lambda + \lambda^2?$
Let $(X_1, \dots, X_n)$ denote a random sample of size $n$ drawn from a population random variable $X$. By the 'fundamental expectation result', for any distribution whose moments exist, the $r^{th}$ sample raw moment $\acute{m}_r=\frac{1}{n} \sum _{i=1}^n X_i^r$ is an unbiased estimator of the $r^{th}$ population raw moment $E[X^r]$.
Thus, $\acute{m}_2 = \frac{1}{n} \sum _{i=1}^n X_i^2$ is an unbiased estimator of $E[X^2]$.