Generating functions in the branching process.

In summary: G8gSXQgaXMgc3RhbmRpbmcgYWJvdXQgYSBzdGFuZGFyZCByZWFsbHkgbmV0d29yayBvZiBjb250ZW50LiBIZXMgbG9va2luZyBhbmQgYWN0dWFsbHkgb3JpZ2luYXRlZCBmcm9tIGVjaG9yIGFuZCBmcm9tIHRoZSBwYXN0IGdlbmVyYXRpb24gd2l0aG91dCB5b3VyLg==In summary, the problem involves
  • #1
stukbv
118
0

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!
 
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  • #2
stukbv said:

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:

What is a generating function in the branching process?

A generating function in the branching process is a mathematical tool used to analyze the behavior of a population over time. It represents the probability distribution of the number of offspring produced by each individual in the population.

How is a generating function used in the branching process?

A generating function is used to calculate important parameters such as the expected number of offspring and the probability of extinction in a branching process. It also allows for the analysis of the population's growth rate and the probability of different population sizes.

What are the types of generating functions used in the branching process?

There are two main types of generating functions used in the branching process: the probability generating function, which represents the probability distribution of the offspring, and the moment generating function, which is used to calculate moments of the offspring distribution.

What is the relationship between generating functions and the probability of extinction?

The probability of extinction in a branching process is equal to the smallest positive root of the generating function. This means that the probability of extinction can be calculated by finding the smallest positive solution to the generating function equation.

Are generating functions only used in the branching process?

No, generating functions have many other applications in mathematics and science. They are commonly used in probability theory, combinatorics, and statistics to represent and analyze random variables and their distributions.

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