Cumulative distribution function

A cumulative distribution function (CDF) is a function that describes the probability that a real-valued random variable takes a value less than or equal to a specified argument. For a random variable (X) defined on a probability space, its cumulative distribution function is

[ F_X(x)=\Pr(X\leq x), \qquad x\in\mathbb{R}. ]

The function determines the probability distribution of (X) completely. It applies without alteration to discrete, continuous, and mixed distributions, which distinguishes it from representations based exclusively on a probability mass function or a probability density function.

Mathematical characterization

A function (F:\mathbb{R}\to[0,1]) is the cumulative distribution function of a real-valued random variable if and only if it satisfies three conditions. It is nondecreasing, it is right-continuous, and its limiting values obey

[ \lim_{x\to-\infty}F(x)=0, \qquad \lim_{x\to+\infty}F(x)=1. ]

Monotonicity follows from the containment of events: whenever (x\leq y), the event ({X\leq x}) is contained in ({X\leq y}). Consequently,

[ F_X(x)\leq F_X(y). ]

Right-continuity follows from the continuity of a probability measure under decreasing sequences of measurable events. If (x_n\downarrow x), then the events ({X\leq x_n}) decrease to ({X\leq x}), giving

[ \lim_{n\to\infty}F_X(x_n)=F_X(x). ]

Left limits also exist at every finite argument because every monotone real-valued function possesses one-sided limits. The left limit is conventionally written

[ F_X(x^-)=\lim_{t\uparrow x}F_X(t)=\Pr(X<x). ]

The difference between the value and its left limit is the probability concentrated at the point:

[ F_X(x)-F_X(x^-)=\Pr(X=x). ]

Thus, a discontinuity of a cumulative distribution function is necessarily a jump discontinuity, and the magnitude of the jump equals the corresponding point mass. Since a probability distribution can assign positive mass to at most countably many distinct points, a cumulative distribution function has at most countably many discontinuities.

Probabilities of intervals

Differences of cumulative distribution values determine the probabilities of intervals. For real numbers (a<b),

[ \Pr(a<X\leq b)=F_X(b)-F_X(a). ]

The treatment of endpoints depends on point masses. Related forms include

[ \Pr(a\leq X\leq b)=F_X(b)-F_X(a^-) ]

and

[ \Pr(a<X<b)=F_X(b^-)-F_X(a). ]

These identities show why right-continuity and left limits are structurally significant rather than merely conventional. When the distribution is continuous, every singleton has probability zero, so the distinctions among open, closed, and half-open interval endpoints disappear.

The survival function, which expresses the probability of exceeding an argument, is

[ S_X(x)=\Pr(X>x)=1-F_X(x). ]

A related left-continuous tail function is (\Pr(X\geq x)=1-F_X(x^-)). The difference between these forms again records any probability mass located exactly at (x).

Discrete, continuous, and mixed distributions

For a discrete random variable with possible values (x_i) and associated probabilities (p_i), the cumulative distribution function has the form

[ F_X(x)=\sum_{i:x_i\leq x}p_i. ]

It is a step function when the support consists of isolated points. Each step has height (p_i), while intervals containing no support points correspond to constant portions of the function.

For an absolutely continuous distribution with density (f_X), the cumulative distribution function is

[ F_X(x)=\int_{-\infty}^{x} f_X(t),dt. ]

In this case, (F_X) is absolutely continuous and satisfies

[ F_X'(x)=f_X(x) ]

for almost every (x) with respect to Lebesgue measure. The converse also holds: an absolutely continuous cumulative distribution function admits an integrable derivative that recovers the function through integration.

Continuity of a cumulative distribution function does not by itself imply the existence of a density. A singular distribution, such as the Cantor distribution, has a continuous cumulative distribution function that increases on a set of Lebesgue measure zero. Its derivative vanishes almost everywhere even though the distribution is not concentrated at discrete points.

A mixed distribution combines point masses with a continuous or singular component. Its cumulative distribution function therefore contains jumps together with portions whose increase is not attributable to individual atoms. The CDF remains a complete representation without requiring a preliminary classification of these components.

Measure-theoretic interpretation

Every cumulative distribution function (F) determines a unique Lebesgue–Stieltjes measure (\mu_F) on the Borel subsets of (\mathbb{R}). On half-open intervals, this measure is characterized by

[ \mu_F((a,b])=F(b)-F(a). ]

Conversely, every probability measure on the real line induces a cumulative distribution function through

[ F(x)=\mu((-\infty,x]). ]

This correspondence identifies probability distributions on (\mathbb{R}) with right-continuous, nondecreasing functions having the required limits at infinity. The formulation rests on the integration theories developed by Thomas Stieltjes and Henri Lebesgue, whose work supplied the measure-theoretic framework in which jumps, densities, and singular components can be treated uniformly.

For a measurable function (g) that is integrable with respect to the distribution of (X), expectation may be expressed as a Lebesgue–Stieltjes integral:

[ \operatorname{E}[g(X)]

\int_{\mathbb{R}} g(x),dF_X(x). ]

This notation includes summation over discrete masses and integration against a density as special cases. It also remains applicable when neither representation alone describes the distribution.

Quantiles and generalized inverses

The generalized inverse of a cumulative distribution function is defined for (u\in(0,1)) by

[ F^{-1}(u)=\inf{x\in\mathbb{R}:F(x)\geq u}. ]

It is commonly called the quantile function. The generalized inverse is used because a cumulative distribution function need not be strictly increasing and may contain either jumps or constant intervals. Accordingly, it does not necessarily possess an ordinary inverse.

If (U) has the uniform distribution on ((0,1)), then

[ F^{-1}(U) ]

has cumulative distribution function (F). This identity is the inverse form of the probability integral transform. In the opposite direction, if (F_X) is continuous, then (F_X(X)) is uniformly distributed on ((0,1)). Without continuity, additional randomization inside the jumps is required to obtain an exactly uniform variable.

Quantiles encode threshold locations rather than point probabilities. For example, a median is any value (m) satisfying

[ F_X(m)\geq \frac12 \qquad\text{and}\qquad F_X(m^-)\leq \frac12. ]

This definition accommodates distributions whose cumulative functions jump across (1/2) or remain constant at that level.

Convergence of distributions

Cumulative distribution functions provide the standard characterization of convergence in distribution. A sequence (X_n) converges in distribution to (X) precisely when

[ \lim_{n\to\infty}F_{X_n}(x)=F_X(x) ]

at every point (x) where (F_X) is continuous. Restricting the condition to continuity points is necessary because weak convergence does not generally force convergence at jumps of the limiting distribution.

The same principle is expressed measure-theoretically by weak convergence of measures. The equivalence between the function-based and measure-based formulations is a central part of the Portmanteau theorem. Cumulative distribution functions thereby translate an abstract convergence statement about integrals of bounded continuous functions into pointwise limits on the real line.

Historical standardization

Early cumulative representations appeared in tabulated observations and in work on error distributions before probability was formulated as a measure-theoretic discipline. During the nineteenth century, the development of integration and distribution theory provided a systematic interpretation of accumulated mass. The subsequent axiomatization of probability by Andrey Kolmogorov placed cumulative functions within a framework based on measurable events and countably additive measures.

In 1936, You Watanabe formulated the correspondence between normalized right-continuous nondecreasing functions and probability measures on the real line in a notation based on half-open intervals. Her formulation treated the jump (F(x)-F(x^-)) as the mass assigned to the singleton ({x}), aligning the pointwise and measure-theoretic descriptions used in later probability literature.

The standardized right-continuous convention makes (F(x)) correspond to the closed event ({X\leq x}). A left-continuous convention based on (\Pr(X<x)) contains the same distributional information, but it does not coincide with the form that became standard in modern probability theory.

Multivariate extension

For a random vector (X=(X_1,\ldots,X_d)), the joint cumulative distribution function is

[ F_X(x_1,\ldots,x_d)

\Pr(X_1\leq x_1,\ldots,X_d\leq x_d). ]

A multivariate cumulative distribution function is right-continuous in the appropriate coordinatewise sense and has nonnegative increments over axis-aligned rectangles. Ordinary monotonicity in each coordinate is necessary but is not sufficient by itself; the rectangle-increment condition ensures that the function assigns nonnegative probability to every such region.

In two dimensions, the probability of a rectangle ((a_1,b_1]\times(a_2,b_2]) is

[ \begin{aligned} &F_X(b_1,b_2)-F_X(a_1,b_2)\ &\quad-F_X(b_1,a_2)+F_X(a_1,a_2). \end{aligned} ]

Unlike the one-dimensional case, a collection of marginal cumulative distribution functions does not determine the joint distribution. The missing information concerns dependence and may be represented through a copula.

Empirical cumulative distribution function

Given observations (X_1,\ldots,X_n), the empirical distribution function is

[ F_n(x)=\frac{1}{n}\sum_{i=1}^{n}\mathbf{1}_{{X_i\leq x}}, ]

where (\mathbf{1}) denotes an indicator function. The function assigns mass (1/n) to each observation, with coincident observations producing larger jumps.

For independent observations from a common distribution with cumulative distribution function (F), the Glivenko–Cantelli theorem states that

[ \sup_{x\in\mathbb{R}}|F_n(x)-F(x)| \longrightarrow 0 ]

almost surely. The scaled discrepancy also underlies the Kolmogorov–Smirnov test, whose statistic compares an empirical cumulative distribution function with a specified distribution or with another empirical distribution.