Understanding that the variance of a sample of normally distributed random variables is chi-squared distributed with mean = population variance, what distribution describes the variance of a sample of poisson distributed random variables.
1$\begingroup$ There is no one distribution or (well-studied) distribution family that answers this and almost surely there is no analytical solution in closed form. (With a sample of size $n,$ the variance estimator multiplied by $n(n-1)$ is a discrete distribution on the non-negative integers with a noticeably irregular shape and many gaps in its support.) Most people would resort to approximation for medium to large samples or large Poisson parameters and some other method (such as simulation) for small samples. $\endgroup$– whuber ♦Mar 24 at 13:29