The Poisson distribution's probability function was derived as a limit of those of binomial distributions. The Poisson mean and variance can also be found from this limit.

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

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

We use the fact that

\[ X \;\; \sim \; \; \lim_{k \to \infty} \BinomDistn\left(k,\; \frac {\lambda} k\right) \]

Denoting a variable with the binomial distribution on the right by \(X_k\), its mean is

\[ E[X_k] \;=\; k \times \frac {\lambda} k = \lambda \]

As \(k \to \infty\), the limiting mean is therefore also \(\lambda\). The binomial variance is

\[ \Var[X_k] \;=\; k \times \frac {\lambda} k \times \left(1 - \frac {\lambda} k \right) \;=\; \lambda \times \left(1 - \frac {\lambda} k \right)\]

As \(k \to \infty\), the limiting variance is \(\lambda\).

Approximate normal distribution

We showed earlier that the Poisson distribution becomes close to normal as its parameter \(\lambda\) increases. We can now be more precise with the parameters of this 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.