## The Galton Watson Process – Part II

The derivation of this equation is not shown here for brevity. If we recursively write out this equation we obtain:

$\displaystyle G_n (s) = G_{n-1} \left( G(s) \right) \\ = G_{n-2} \left( G (G(s)) \right) \\ = G_{n-3} \left( G (G (G(s))) \right) \\ = \dots \\ = G_{1} \left( G (G (G(\dots (s)\dots ) )) \right) \\ = G \left( G (G (G(\dots (s)\dots ) )) \right) \ \ \ \ \ (15)$

From this equation we see:

$\displaystyle G_n = G \left( G (G (G(\dots (s)\dots ) )) \right) \ \ \ \ \ (16)$

Then ${G_{n-1}}$ is simply ${G_{n}}$ that is one less recursive function of ${G}$, i.e.

$\displaystyle G_{n-1} = G (G (G(\dots (s)\dots ) )) \ \ \ \ \ (17)$

Then by plugging Eq.17 into Eq.16 we obtain:

$G_n \left( s \right) = G\left( G_{n-1} \left(s\right) \right) \ \ \ \ \ \mbox {for}\, n = 1,2,3, \dots (18)$

which is much more handy for calculating the probability of extinction than the previous form, $G_n (s) = G_{n-1} \left( G(s) \right)$. This relation is useful if we realize that the probability of extinction is identical to the probability generation function when s=0. This is because $G_n(0) = P(X_n=0)$, where $P(X_n=0)$ is the probability that the total number of people of the n-th generation $(X_n)$ is equal to zero. Thus the linkage between the probability of extinction and the probability generation function is:

$e_n = G_n \left(0 \right) = P \left(X_n = 0\right) \, \, \, \, \,(19)$