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## Homework Statement

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!