Question 1184789
The proof for {{{Var(Y) = lambda}}} if Y is the Poission distribution can be found in many standard texts in probability and statistics, and even in the internet.


We show {{{E(Y(Y-1)) = lambda}}}.


{{{E(Y(Y-1)) = E(Y^2) - E(Y) = (sigma^2 + mu^2) - mu = lambda + lambda^2-lambda = lambda^2}}}, 


since {{{E(Y^2) = sigma^2 + mu^2}}} and {{{sigma^2 = mu = lambda}}} for the Poisson distribution.