Mean and Variance

Figure 1: Poisson distribution with mean value 5

The Poisson distribution for $\bar k = 5$ is shown in Fig. 1. Notice that it peaks at $k = 5$. Let us determine the mean and variance for the Poisson distribution. The mean is just

$$\langle k\rangle = \sum_{k=0}^{\infty} k P(k,\bar k).$$ (15)

A little algebra gives

$$\langle k\rangle = \bar k.$$ (16)

This result is naturally what we would expect, of course. The variance is given by

$${\rm Var}(k) = \langle k^{2}\rangle - \langle k\rangle^{2} = \sum_{k=0}^{\infty}k^{2}P(k,\bar k) - \bar k^{2}$$ (17)

A little algebra shows that the first term is just $\bar k(\bar k + 1)$ so

$${\rm Var}(k) = \bar k.$$ (18)

This result says that the standard deviation is approximately $\sqrt{\bar k}$. Actually we have to be careful about using the term ``standard deviation'' for the Poisson distribution, unless $\bar k$ is large. For small $\bar k$ the shape is not very much like a Gaussian, but for large $\bar k$ the shape approximates a Gaussian reasonably well.