2.4 Distribution function

Knowing that $\Omega$ is the set of all possible experiment outcomes, we often want to know ``if I do an experiment, what is the probability that the outcome is $\omega$ (lowercase omega)?'' This is the probability distribution function:

${\rm Pr}(X = u)$

In this notation, $X$ is a random variable, $u$ is a numeric value. Consequently, ${\rm Pr}(X = u)$ is the probability that the outcome of an experiment maps to the value $u$ using the random variable function.

Note that $\sum_{u} {\rm Pr}(X = u) = 1$ , which means the sum of all probabilities over the entire range of $X$ equals one.