Random variable

A random variable has a set of possible values, each with some probability. The sum of all the probabilities is 1.

A random variable can be defined by its probability distribution function (PDF). For some random variable $x$, $p(x_0) \d x$ denotes the probability that $x \in [x_0, x_0 + \dx]$. For a discrete random variable, the PDF can be defined in terms of the Dirac delta function.

For example, let $x$ be the result of a coin flip ($-1$ heads, $+1$ tails). It’s PDF can be written $p(x) = \frac 12 \delta(x-1) + \frac 12 \delta(x+1)$.

A random variable can also be described by it’s cumulative distribution function (CDF, a.k.a. the cumulant). The CDF is simply the integral of the PDF, and represents the probability of finding the random variable to be less than some value.