3 Poisson Distribution

The Poisson distribution is given by the equation:

(6)

The Poisson distribution is useful as a description of the number of children that are produced from parents in the Galton-Watson process. This is because it has the property that the expected value is equal to . This makes the interpretation of the function easy: if we want a families to have, on average, two kids, we would set .

To generate a useful description using the Poisson distribution we must first determine in a closed form. This is done by plugging Eq.6 into Eq.2, producing the following generating function:

(7)

The part of the equation above that is in brackets is an exponential series expansion of . Thus replacing the part in brackets we obtain:

(8)

In my next post I will continue this discussion by looking at the **probability of extinction**.