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.
Xn is the number of particles in the nth generation
The distribution of a typical family size is Z and so Xn is the sum of appropriate Zi's
I need a generating function of the number Xn in the nth generation.
2. The attempt at a solution
I know that Fn+1(s) = E [sXn+1]
from the definition of generating functions and how to derive them.
But my lecturer then goes on to say that = ƩE[sXn+1|Xn=j] * P[Xn=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!