# Probability of at least n particles Galton–Watson process [closed]

The process starts with k particles with distribution below

0 p0 1 p1 2 p2 3 p3 4 p4 5 p5

0< E < 1

$$G(z)$$ is p.g.f for the offspring distribution

$$G(z) = p0 + p1*z + p2*z^2+p3*z^3+p4*z^4+p5*z^5$$

$$Q(z)$$ is p.g.f of the total amount of particles

$$Q(z) = z*G(Q)$$

$$Q = z*(p0+p1*Q+p2*Q^2+p3*Q^3+p4*Q^4+p5*Q^5)$$

How can I find the probability that there would be at least n particles in this process?

## closed as off-topic by RP_, Chris Godsil, LSpice, ofer zeitouni, Mark WildonMay 18 at 15:32

This question appears to be off-topic. The users who voted to close gave these specific reasons:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – RP_, LSpice, Mark Wildon
• "MathOverflow is for mathematicians to ask each other questions about their research. See Math.StackExchange to ask general questions in mathematics." – Chris Godsil, ofer zeitouni
If this question can be reworded to fit the rules in the help center, please edit the question.

• There is no miracle formula for the exact answer as far as I know. You just have to compute the probabilities honestly or express them in terms of $Q$ as integrals using the Cauchy formula. If you want just to approximate, then tell in what regime. – fedja May 16 at 6:26