Mean and Variance of a Poisson

Recall the moment generating function of the Poisson distribution.

Both the first and second cumulants of the Poisson distribution equal its rate parameter.

1. 1. State the mean

   For $X\sim \text{Poisson}(\lambda )$, $E[X]=\lambda$.

2. 2. State the variance

   The Poisson distribution has the special property that its variance also equals $\lambda$.

   $$\text{Var}(X)=\lambda$$

Answer: Mean = lambda, variance = lambda