Cumulative distribution function

The cumulative distribution function (CDF) of a random variable X is the function F_X(x) = P(X ≤ x), giving the probability that X takes a value less than or equal to x. Unlike the probability density function, which may not exist for all random variables, the CDF always exists and completely characterizes the variable's distribution.

The CDF encodes all probabilistic information about a random variable in a single function. It is right-continuous, non-decreasing, and bounded between 0 and 1. The step discontinuities in a CDF reveal the discrete components of a distribution; its continuous regions reveal the density. The CDF is the fundamental object; the density is merely its derivative where one exists. This inversion of pedagogical priority — teaching density first, CDF second — is one of statistics' small tragedies. Students learn to differentiate before they learn what they are differentiating.