Notes - Probability MT22, Probability generating functions

What is the probability generating function $G _ X(s)$ for $X$ where $\text{Im } X \subset \{ 0, 1, 2, … \}$?

\[G _ X(s) = \sum^{\infty} _ {k = 0} \mathbb{P}(X = k)s^k\]

How can you write $G _ X(s) = \sum^{\infty} _ {k = 0} \mathbb{P}(X = k)s^k$ as an expectation?

\[G _ X(s) = \mathbb{E}[s^X]\]

When is $G _ X(s) = \sum^{\infty} _ {k = 0} \mathbb{P}(X = k)s^k$ guarenteed to converge since the probabilities must sum to one?

\[\vert s \vert \le 1\]

How can you recover $\mathbb{P}(X = k)$ from $G _ X(s)$?

\[\frac{G _ x^{(k)}(0)}{k!}\]

How can you determine $\mathbb{E}[X]$ from $G _ X(s)$?

\[\mathbb{E}[X] = G _ X'(1)\]

What is $G _ {X+Y}(s)$ for independent $X$ and $Y$?

\[G _ X(s) G _ Y(s)\]

How can you determine $\text{var } X$ from $G _ X(s)$?

\[\text{var } X = G'' _ X(1) + G' _ X(1) - (G' _ X(1))^2\]

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.

@Prove that if $X$ is a random variable, then the distribution of $X$ is uniquely determined by its probability generating function $G _ X$.

@todo (probability, page 36).

@important~

@Prove that if $X$ and $Y$ are independent random variables, then

\[G _ {X+Y}(s) = G _ X(s) G _ Y(s)\]

@todo (probability, page 37).

Probability MT22, Probability generating functions