The Poisson distribution's mean and variance can also be found as a limit of the binomial distribution's mean and variance.

Mean and variance

If \(X\) has a \(\PoissonDistn(\lambda)\) distribution with probability function

\[ p(x) \;\;=\;\; \frac {\lambda^x e^{-\lambda}} {x!} \quad\quad \text{ for } x=0, 1, \dots \]

then its mean and variance are

\[ E[X] \;=\; \lambda \spaced{and} \Var[X] \;=\; \lambda \]

(Proved in full version)

Approximate normal distribution

We can now be more precise about the parameters of the normal approximation to the Poisson distribution.

\[ \PoissonDistn(\lambda) \;\;\xrightarrow[\lambda \to \infty]{} \;\; \NormalDistn(\mu = \lambda,\; \sigma^2 = \lambda) \]

This approximation is reasonably good even when \(\lambda\) is as low as 20.