# Offspring generating function in Branching process.

Find the extinction probability for a branching process with offspring distribution $$a =(1∕6, 1∕2, 1∕3)$$.

Solution

The mean of the offspring distribution is
$$\mu = 0(1∕6)+ 1(1∕2)+ 2(1∕3)= 7∕6 > 1$$,
so this is the supercritical case.

The offspring generating function is $$G(s)= 1/6 + s/2 + s^2/3$$.

Solving $$s = G(s)= 1/6 + s/2 + s^2/3$$ gives the quadratic equation $$s^2∕3 − s∕2 + 1∕6 = 0$$, with roots $$s = 1$$ and $$s = 1∕2$$.

The smallest positive root is the probability of eventual extinction $$e = 1∕2$$.

I have several questions.

1. What would be the offspring generating function in critical and sub-critical cases?
2. Why is $$s = G(s)$$?
• What do you know about Branching processes? The second property is a basic theorem on extinction probability. The generating function of a critical or sub-critical BP depends on the offsping distribution , so what type of answer can be given to 1)? – Kavi Rama Murthy May 11 at 0:10
• @KaviRamaMurthy, The generating function of a critical or sub-critical BP depends on the offspring distribution - I want to see what would PGF look like in each of those cases. – user366312 May 11 at 0:31
• @KaviRamaMurthy, The second property is a basic theorem on extinction probability. - kindly, supply me some material to study. – user366312 May 11 at 0:32
• @KaviRamaMurthy, What do you know about Branching processes? - I just started it today. – user366312 May 11 at 0:32