Notes - Probability MT22, Random sums theorem
Can you state the random sums theorem in full?
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
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 $G _ N(G _ X(s))$.
What expression do you start with that the proof follows from?
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 $G _ N(G _ X(s))$.
\[\mathbb{E}[s^{X_1+X_2+\ldots+X_N}]\]