Cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable \(X\), or just distribution function of \(X\), evaluated at \(x\), is the probability that \(X\) will take a value less than or equal to \(x\). It fully describes the probability distribution of a random variable and is one of the most fundamental concepts in the field. For every real number \(x\), the CDF is given by
\[
F_X(x) = \operatorname{P}(X \le x),
\]
where the right-hand side represents the probability of the event \(X \le x\). The CDF is a universal way to specify the distribution of any real-valued random variable, regardless of whether it is discrete, continuous, or a mixture of the two. In addition to its central role in univariate probability theory, the concept generalizes naturally to multivariate random vectors and to more abstract settings, such as random elements in metric spaces.

Definition and basic notation

Let \(X\) be a real-valued random variable defined on a probability space \((\Omega, \mathcal{F}, P)\). The cumulative distribution function of \(X\) is the function \(F_X : \mathbb{R} \to [0,1]\) defined by
\[
F_X(x) = P(\{\omega \in \Omega \mid X(\omega) \le x\}) = P(X \le x).
\]
The subscript \(X\) is often dropped when the associated random variable is clear from context, and one simply writes \(F(x)\). The CDF is sometimes referred to as the distribution function or, in older literature, the cumulative frequency function.

Every CDF possesses the following characteristic properties:

1. Boundedness: \(\lim_{x \to -\infty} F(x) = 0\) and \(\lim_{x \to +\infty} F(x) = 1\).
2. Monotonicity: If \(x_1 \le x_2\), then \(F(x_1) \le F(x_2)\).
3. Right-continuity: For every real number \(a\), \(\lim_{x \to a^+} F(x) = F(a)\).

These three properties are not only necessary but also sufficient for a function \(F: \mathbb{R} \to [0,1]\) to be a valid cumulative distribution function. That is, any function satisfying the three conditions is the CDF of some random variable (the Lebesgue–Stieltjes measure induced by \(F\) can be used to construct the distribution). Because of this characterization, the CDF provides a complete description of the probability law of a random variable.

Relationship with the probability density and mass functions

For a continuous random variable, the CDF is the integral of its probability density function (PDF). If \(X\) has density \(f_X\), then
\[
F_X(x) = \int_{-\infty}^{x} f_X(t) \, dt.
\]
By the fundamental theorem of calculus, the density is the derivative of the CDF wherever the derivative exists: \(f_X(x) = F'_X(x)\) at all points of continuity of \(f_X\). For an absolutely continuous random variable, this relationship holds almost everywhere.

For a discrete random variable taking values \(x_1, x_2, \dots\) with probability mass function \(p_X\), the CDF is a step function:
\[
F_X(x) = \sum_{x_i \le x} p_X(x_i).
\]
Consequently, the CDF of a discrete variable is discontinuous at each value that has positive probability, with a jump size equal to that probability.

Mixed distributions, which have both continuous and discrete components, also have CDFs that are a combination of continuous and step features.

Properties

Complement and survival function

The function \(S_X(x) = 1 - F_X(x) = P(X > x)\) is called the survival function or reliability function. It is especially important in survival analysis, reliability engineering, and actuarial science. The survival function is non-increasing, left-continuous (as opposed to the right-continuous CDF), and tends to 1 as \(x \to -\infty\) and to 0 as \(x \to +\infty\).

Quantile function

The quantile function (or inverse CDF) is the generalized inverse of the CDF. For a distribution function \(F\), the quantile function \(Q : (0,1) \to \mathbb{R}\) is defined by
\[
Q(p) = \inf\{x \in \mathbb{R} : p \le F(x)\}.
\]
For continuous and strictly increasing CDFs, the quantile function coincides with the ordinary inverse \(F^{-1}\). The quantile function is used to generate random samples via the inverse transform sampling method and to define important descriptive measures such as the median, quartiles, and percentiles.

Empirical cumulative distribution function

Given a sample of \(n\) independent and identically distributed observations \(x_1, x_2, \dots, x_n\), the empirical cumulative distribution function (eCDF) \(F_n\) is defined as
\[
F_n(x) = \frac{1}{n} \sum_{i=1}^{n} \mathbf{1}_{[x_i \le x]},
\]
where \(\mathbf{1}\) is the indicator function. The eCDF is a step function that jumps by \(1/n\) at each data point. By the Glivenko–Cantelli theorem, the empirical CDF converges uniformly to the true CDF almost surely as the sample size increases. It is a fundamental tool in nonparametric statistics and goodness-of-fit tests, such as the Kolmogorov–Smirnov test.

Right-continuity and left-hand limits

By convention, the CDF is defined with a non-strict inequality, which makes it right-continuous. Some treatments (especially in Eastern European literature) define the CDF as \(P(X < x)\), which yields a left-continuous function. The choice is largely a matter of convention, but the right-continuous version is standard in most modern probability and statistics texts. The left-hand limit \(F(x^-) = \lim_{t \uparrow x} F(t) = P(X < x)\) is always defined and equals \(F(x)\) if and only if \(P(X = x) = 0\). The difference \(F(x) - F(x^-) = P(X = x)\) gives the probability mass at \(x\), if any.

Multivariate cumulative distribution function

For a random vector \(\mathbf{X} = (X_1, \dots, X_d)\) taking values in \(\mathbb{R}^d\), the joint cumulative distribution function is defined by
\[
F_{\mathbf{X}}(x_1, \dots, x_d) = P(X_1 \le x_1, \dots, X_d \le x_d).
\]
The multivariate CDF has properties analogous to the univariate case: it is continuous from the right in each coordinate, tends to 0 as any argument approaches \(-\infty\), tends to 1 as all arguments approach \(+\infty\), and satisfies certain non-negativity conditions on rectangle probabilities (it is "\(d\)-increasing"). However, in multivariate settings the CDF is often less convenient than the joint density or the characteristic function, and its use is more limited except in copula theory, where marginal CDFs are transformed to uniform variables.

Copulas and dependence

Sklar’s theorem states that any multivariate CDF can be expressed in terms of its marginal distributions and a copula, which captures the dependence structure. Specifically, if \(F\) is a joint CDF with margins \(F_1, \dots, F_d\), then there exists a copula \(C : [0,1]^d \to [0,1]\) such that
\[
F(x_1, \dots, x_d) = C\bigl(F_1(x_1), \dots, F_d(x_d)\bigr).
\]
This decomposition is fundamental in modern risk management, finance, and multivariate statistics.

Examples

Continuous distributions

• Normal distribution: The CDF of a standard normal random variable is

\[
\Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} e^{-t^2/2} \, dt.
\]
It has no closed-form expression in terms of elementary functions but is widely tabulated and available as a standard function in statistical software.

• Exponential distribution: For rate parameter \(\lambda > 0\),

\[
F(x) = \begin{cases}
1 - e^{-\lambda x}, & x \ge 0,\\
0, & x < 0.
\end{cases}
\]

• Uniform distribution on \([a,b]\):

\[
F(x) = \begin{cases}
0, & x < a,\\
\dfrac{x-a}{b-a}, & a \le x \le b,\\
1, & x > b.
\end{cases}
\]

Discrete distributions

• Bernoulli distribution: For a success probability \(p\),

\[
F(x) = \begin{cases}
0, & x < 0,\\
1-p, & 0 \le x < 1,\\
1, & x \ge 1.
\end{cases}
\]

• Poisson distribution: With mean \(\lambda\),

\[
F(x) = e^{-\lambda} \sum_{k=0}^{\lfloor x \rfloor} \frac{\lambda^k}{k!},
\]
where \(\lfloor x \rfloor\) denotes the floor of \(x\).

Mixed distribution example

Consider a random variable that equals 0 with probability 0.5 and otherwise follows an exponential distribution with rate 1. Its CDF is
\[
F(x) = \begin{cases}
0, & x < 0,\\
0.5 + 0.5(1 - e^{-x}) = 0.5(1 + e^{-x}), & x \ge 0.
\end{cases}
\]
Note the jump at \(x = 0\) of size 0.5.

Applications

The cumulative distribution function is a core concept that appears in virtually every branch of probability and statistics. Some prominent applications include:

• Hypothesis testing: The \(p\)-value is often defined as a transformation of a test statistic using its CDF under the null hypothesis. For example, in one-sided tests, \(p = 1 - F(T)\) (or \(F(T)\) for left-tailed tests).
• Confidence intervals: Critical values are obtained from quantile functions (inverses of CDFs) of standard distributions such as the normal, \(t\), or chi-squared.
• Random variate generation: The inverse transform method uses the quantile function to convert uniform random numbers into samples from any distribution.
• Model diagnostics: The empirical CDF is used in probability plots (e.g., Q–Q plots) to assess the fit of a parametric distribution to data.
• Risk management: In finance and insurance, the value-at-risk (VaR) is a quantile of the loss distribution, directly derived from the CDF. The expected shortfall or conditional value-at-risk also depends on the CDF.
• Nonparametric estimation: The empirical CDF is the basis for many nonparametric procedures, including the Kolmogorov–Smirnov test, Cramér–von Mises test, and bootstrap methods.
• Information theory: The CDF transform (probability integral transform) maps any continuous random variable to a uniform distribution on \([0,1]\), which is a key property for copula construction and distribution-free goodness-of-fit tests.

The cumulative distribution function stands as a central unifying tool because it completely determines the stochastic behavior of a random variable and connects the abstract measure-theoretic foundation of probability to practical computations and visualizations.