Random Variable

A random variable on $S$ is a measurement of a random outcome $s\in S$.

The probability that a random variable $X$ takes the value $k\in \mathbb{R}$ is given by:

$$\text{Prob}(X=k)=\text{Prob}(\{s\in S:X(s)=k\})$$

It follows similarly that:

$$\begin{array}{c}\text{Prob}(X\le k)=\text{Prob}(\{s\in S:X(s)\le k\})\\ \text{Prob}(X\ge k)=\text{Prob}(\{s\in S:X(s)\ge k\})\end{array}$$