Generating functions in the branching process.

[stukbv]

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^X_n+1]
from the definition of generating functions and how to derive them.

But my lecturer then goes on to say that = ƩE[s^X_n+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!

 

[Ray Vickson]

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^X_n+1]
from the definition of generating functions and how to derive them.

But my lecturer then goes on to say that = ƩE[s^X_n+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!

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

RGV