Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have a branching process of the form $p_0=0.1$, $p_1 = 0.6$, $p_2 = 0.3$. (any other $p_n = 0$). $Z_0$, the original population is $1$. $Z_1$ is the population after $1$ timestep, $Z_2$ is the population after $2$ timesteps, etc.

How can I compute the probability mass function of $Z_3$? It would suffice to have an expression I could expand on my TI-89.

share|cite|improve this question
What are $p_0$, $p_1$, and $p_2$ the probabilities of? – Rahul Nov 4 '10 at 1:50
Probabilities of generating 0, 1, 2 children respectively. – user2617 Nov 4 '10 at 1:53
Why do you have to use a TI-89? – Yuval Filmus Nov 4 '10 at 1:57
up vote 3 down vote accepted

The probability of having $k$ individuals after 3 steps is the coefficient of $x^k$ in the expansion of $f(f(f(x)))$, where $f(x) = 0.1 + 0.6x + 0.3x^2$.

This generalizes a lot: you can have different types of individuals, each with their own branching probabilities, which can change from step to step.

share|cite|improve this answer
More precisely this is the generating function. – PEV Nov 4 '10 at 2:07

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.