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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?

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

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  • $\begingroup$ 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. $\endgroup$ – fedja May 16 at 6:26

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