Lecture - Probability MT22, IX
Flashcards
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?
\[|s| \le 1\]
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 $G _ X(s)$?
\[\text{var } X = G''_X(1) + G'_X(1) - (G'_X(1))^2\]