# Lecture - Probability MT22, XI

### Flashcards

What does the uniqueness theorem for probability generating function imply?

If you can show that the PGF of a random variable is the same as the PGF of a known distribution, that variable must be distributed the same.

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 jist 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 $s = G(s)$