## The Galton Watson Process – Part I

3 Poisson Distribution

The Poisson distribution is given by the equation:

$P(X = i) = \frac{\lambda^i}{i!} e^{-\lambda}$          (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 $\lambda$. This makes the interpretation of the function easy: if we want a families to have, on average, two kids, we would set $\lambda=2$.

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

$G(s) = \frac{\lambda^0}{0!} e^{-\lambda} s^0 + \frac{\lambda^1}{1!} e^{-\lambda} s^1 + \frac{\lambda^2}{2!} e^{-\lambda} s^2 + \frac{\lambda^3}{3!} e^{-\lambda} s^3 + \dots \\$         (7)

$= \left[\frac{\left( \lambda s\right)^1}{1!} + \frac{\left( \lambda s\right)^2}{2!} + \frac{\left( \lambda s\right)^3}{3!} + \dots \right] e^{-\lambda}$

The part of the equation above that is in brackets is an exponential series expansion of $e^{\lambda s}$. Thus replacing the part in brackets we obtain:

$G(s) = e^{\lambda s} e^{-\lambda} = e^{\lambda (s - 1)}$          (8)

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