Random variable

A random variable on a probability space $\Omega$ is just a function with domain $\Omega$. Rather than writing a random variable as $f(\omega)$ everywhere, the convention is to write a random variable as a capital letter ($X, Y , S$, etc.) and make the argument implicit: $X$ is really $X(\omega)$. Variables that are not random (or are not variable) are written in lowercase.

For example, consider the probability space corresponding to rolling two independent fair six-sided dice. There are $36$ possible outcomes in this space, corresponding to the $6\times6$ pairs of values $\langle x, y\rangle$ we might see on the two dice. We could represent the value of each die as a random variable $X$ or $Y$ given by $X(\langle x, y\rangle) = x$ or $Y(\langle x, y\rangle) = y$, but for many applications, we do not care so much about the specific values on each die. Instead, we want to know the sum $S = X + Y$ of the dice. This value $S$ is also random variable; as a function on $\Omega$, it is defined by $S(\langle x, y\rangle) = x + y$.