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Generating functions in the branching process.

  • Thread starter stukbv
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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.

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!

Thank you!
 

Answers and Replies

  • #2
Ray Vickson
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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.

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!

Thank you!
It's just a standard result in Probability. Suppose [itex] \{A_k \}[/itex] is a partition of the sample space [itex] \Omega [/itex], meaning that the A_k are disjoint and their union is Ω. Then, for any discrete random variable B we have [tex] \Pr \{B=j\} = \sum_k \Pr\{B=j|A_k\} \Pr \{A_k\}.[/tex] Thus, for any f >= 0 we have
[tex] E f(B) = \sum_j f(j) \Pr\{B=j \} = \sum_k \Pr\{A_k\} \sum_j f(j) \Pr\{B=j|A_k\}
=\sum_k E[f(B)|A_k] \Pr\{A_k\}, [/tex]
where I have swapped the order of summation, which is OK for a positive function.

RGV
 
Last edited:

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