Do the mean and variance of a random variable define it completely? In particular,
if $X$ is a random variable, such that $E(X) = \operatorname{Var}(X) = u$,
does that imply that $X \sim \operatorname{Poisson}(u)$ ?
If so, then a proof that a RV is Poisson would require only to show the above relationship which is very useful. Otherwise, what measures can uniquely define a random variable, or a Poisson Random Variable?