Probability MT22, Random sums theorem
Flashcards
If $X _ i \sim \mathrm{Poi}(\lambda _ i)$, and all $X _ i$ are mutually independent, what is $G _ {\sum^n _ {i=1} X _ i}(s)$?
@State the random sums theorem.
Let $X _ 1, X _ 2, \ldots$ be i.i.d. non-negative integer-valued random variables with p.g.f. $G _ X(s)$. Let $N$ be another non-negative integer-valued random variable, independent of $X _ 1, X _ 2, \ldots$ and with p.g.f. $G _ N(s)$.
Then the p.g.f. of $\sum^N _ {i=1} X _ i$ is $G _ N(G _ X(s))$.
When proving the random sums theorem (∆random-sums-theorem), what expression do you start with that the proof follows from?
What are the conditions on the $X _ i$s in the random sums theorem about $R = \sum^N _ {i=1} X _ i$?
They are identically distributed.
What are the conditions on the $X _ i$s and $N$ in the random sums theorem about $R = \sum^N _ {i=1} X _ i$?
They are independent.
Let $X _ i, i \ge 1$ be i.i.d. random variables with p.g.f. $G _ X(s)$ and let $N$ be another random variable independent of all $X _ i$. What is the p.g.f. of $R = \sum^N _ {i=1} X _ i$?
@Prove the random sums theorem (∆random-sums-theorem): for $X _ 1, X _ 2, \ldots$ i.i.d. non-negative integer-valued random variables with p.g.f. $G _ X(s)$ and $N$ a non-negative integer-valued random variable independent of the $X _ i$ with p.g.f. $G _ N(s)$, the p.g.f. of $\sum^N _ {i=1} X _ i$ is $G _ N(G _ X(s))$.
@todo (probability, page 41).