# Lecture - Probability MT22, IV

### Flashcards

What does it mean for a family of events $\{ A _ i, i \in I \}$ to be independent?

For all finite subsets $J$ of $I$

\[\mathbb{P}\left(\bigcup_{i \in J} A_i \right) = \prod_{i \in J} \mathbb{P(A_i)}\]What sets is a discrete random variable $X$ a function on?

\[\Omega \to \mathbb{R}\]

What is the first condition for a function $X : \Omega \to \mathbb{R}$ to be a discrete random variable, in English (it’s about images)?

The image of $X$ is a countable set.

What is the second condition for a function $X : \Omega \to \mathbb{R}$ to be a discrete random variable (about what it means for $X = x$)?

\[\\{\omega \in \Omega : X(\omega) = x\\} \in \mathcal{F}\]

Can you expand “the image of a discrete random variable $X$” into notation?

\[\\{X(\omega) : \omega \in \Omega\\}\]

What is the probability mass function of a discrete random variable $X$?

\[p_X(x) = \mathbb{P}(X = x)\]