Probability MT22, Branching processes
Flashcards
What’s the “offspring distribution” in a branching process?
The distribution for the number of children of each individual.
What assumption is made about individuals reproducing in a branching process, other than having the same offspring distribution?
They reproduce independently.
If the offspring distribution is $G(s)$, what’s the distribution for the number of individuals in the $n$-th generation?
repeated $n$ times.
What’s the basic gist of proving the fact $G _ {n+1}(s) = G(G _ n(s))$ for a branching process?
The random sums theorem.
If each individual in a generation of a branching process gives birth to $\mu$ children, what’s the expected number of children in the $n$-th generation?
What’s the probability that a branching process with generating function $G(s)$ dies out?
Smallest solution to $s = G(s)$
@Prove that if a branching process has offspring distribution given by probability generating function $G$ then the distribution for the number of individuals at generation $n$ is given by
\[G^n(s)\]
@todo (probability, page 42).
@important~
Let $X _ n$ be the number of children in the $n$-th generation of a branching process and that the mean number of children of a single individual is $\mu$. @Prove that
\[\mathbb E[X _ n] = \mu^n\]
@todo