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.

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