(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I am told that I have particles which each yield a random number of offspring of known distribution independently from each other and from the past generations.

X_{n}is the number of particles in the nth generation

The distribution of a typical family size is Z and so X_{n}is the sum of appropriate Z_{i}'s

I need a generating function of the number X_{n}in the nth generation.

2. The attempt at a solution

I know that F_{n+1}(s) = E [s^{Xn+1}]

from the definition of generating functions and how to derive them.

But my lecturer then goes on to say that = ƩE[s^{Xn+1}|X_{n}=j] * P[X_{n}=j ]

Summed over j.

How does he get from one to the other? If I can make this link then I can go on to show what I need to!

Thank you!

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# Generating functions in the branching process.

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