Random Variables and Expectation

A random variable is a function from outcomes to numbers: $X: \Omega \to \mathbb{R}$. Roll two dice, let $X$ = sum: $X$ maps $(3,4) \mapsto 7$.

Discrete variables have a PMF $p(x) = P(X = x)$; continuous ones a density (next lessons). The CDF $F(x) = P(X \le x)$ works for both and fully characterizes the distribution.