Random Variables

Random variables are event spaces that are ordinal. They can be thought of as an $\Omega$ event space, just as in basic probability. They differ in that the set of events all correspond to points on the real number line. This allows one to ask questions about ordered values. For example, it becomes coherent to ask what the probability is of some random variable being greater than some value. The probability for some event is a function dependent on that event; in this way, we can leverage all the power of functional mathematics, including calculus, in analyzing chance events.

Any random variable must be defined with a probability distribution to be coherent. If we say that $X$ is a random variable, then $X$ necessarily has some associated probability function for all values that it can take on.