The cumulative distribution function (abbreviated cdf) describes the probability distribution of a real-valued random variable, X, completely. For every real number x, the cdf is given by
As an example, suppose X is uniformly distributed on the unit interval [0,1]. Then the cdf is given by
Every cumulative distribution function F is monotone increasing and continuous from the right. Furthermore, we have lim_{x→-∞} F(x) = 0 and lim_{x→∞} F(x) = 1. Every function with these four properties is a cdf.
If X is a discrete random variable, then it attains values x_{1}, x_{2}, ... with probability p_{1}, p_{2} etc., and the cdf of X will be discontinuous at the points x_{i} and constant in between.
If the cdf F of X is continuous, then X is a continuous random variable; if furthermore F is absolutely continuous, then there exists a Lebesgue-integrable function f(x) such that
The Kolmogorov-Smirnov test is based on cumulative distribution functions and can be used to test to see whether two empirical distributions are different or whether an empirical distribution is different from an ideal distribution. The closely related Kuiper's test (pronounced in Dutch the way an Cowper might be pronounced in English) is useful if the domain of the distribution is cyclic as in day of the week. For instance we might use Kuiper's test to see if the number of tornadoes varies during the year or if sales of a product vary by day of the week or day of the month.
See also Descriptive statistics, Probability distribution.