Example 3: Poisson distribution

Often we do not know the total number of trials, but we just know that events occur independently and not simultaneously at a mean rate of $\mu$ in a certain region of space or in an interval time. For example, we might know that there are a mean number of 20 hurricanes in the Atlantic region per year. In such cases, the number of events $X$ that occur in a fixed region or time interval is given by the Poisson distribution $X\sim Poisson(\mu)$ defined by

$$\Pr(X=m) = \frac{{\rm e}^{-\mu}\mu^m}{m!}=f(m;\mu)$$ (4.3)

for $m=0,1,\ldots$. A Poisson distributed count variable has expectation $E(X)=\mu$ and variance $Var(X)=\mu$. The Poisson distribution approximates the Binomial distribution in the limit of large $n$ and finite $\mu=n\pi$. The sum of two independent Poisson distributed variables is also Poisson distributed $X_1+X_2\sim Poisson(\mu_1+\mu_2)$. Meteorological events such as storms often satisfy the independence and non-simultaneity criteria necessary for a Poisson process and so the number of such events in a specified region or time interval can be satisfactorily modelled using the Poisson distribution.