Lecture - Probability MT22, XI

Created: November 19, 2022

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?
\[G(G(G(G(G\ldots(s)))))\]

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?
\[\mu^n\]

What’s the probability that a branching process with generating function $G(s)$ dies out?

Smallest solution to $s = G(s)$

What does it mean if $(A _ n) _ {n\ge1}$ is an increasing family of events?
\[A_n \subseteq A_{n+1}\]

If $(A _ n) _ {n \ge 1}$ is an increasing family of events, what is the expression for $\lim\ _ {n \to \infty} \mathbb{P}(A _ n)$ when the random variable is discrete?
\[\mathbb{P}\left(\bigcup_{n=1}^\infty A_n\right)\]

If $(A _ n) _ {n \ge 1}$ is an increasing family of events, what is the expression for $\mathbb{P}\left(\bigcup _ {n=1}^\infty A _ n\right)$ when the random variable is discrete?
\[\lim_{n \to \infty} \mathbb{P}(A_n)\]


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