Continuous random variable

A continuous random variable is a random variable whose probability law assigns no positive probability to any individual value. Under a narrower convention common in mathematical statistics, the term denotes a random variable whose distribution is absolutely continuous with respect to Lebesgue measure, so that probabilities can be represented by integrating a probability density function. The distinction between these conventions is material because a distribution can have a continuous cumulative distribution function without possessing a density.

Continuous random variables provide mathematical models for quantities represented on continua, including elapsed time and spatial position. Their defining properties concern probability measures rather than the physical character of the quantities being modeled. A continuously measured quantity can therefore be represented by a discrete random variable after rounding, while a countable physical system can be assigned a continuous parameter in an abstract model.

Measure-theoretic definition

Let ((\Omega,\mathcal F,\mathbb P)) be a probability space. A real-valued random variable is a measurable function

[ X:\Omega\longrightarrow\mathbb R. ]

The distribution of (X) is the pushforward measure

[ \mu_X(B)=\mathbb P(X\in B) ]

defined for every Borel set (B\subseteq\mathbb R). The variable has no atoms when

[ \mathbb P(X=x)=\mu_X({x})=0 ]

for every (x\in\mathbb R). This condition is equivalent to continuity of the cumulative distribution function

[ F_X(x)=\mathbb P(X\leq x). ]

An absolutely continuous random variable satisfies the stronger condition

[ \mu_X\ll\lambda, ]

where (\lambda) denotes Lebesgue measure. The Radon–Nikodym derivative

[ f_X=\frac{d\mu_X}{d\lambda} ]

is then a probability density function, and

[ \mathbb P(X\in B)=\int_B f_X(x),dx. ]

Consequently,

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

The density is determined only up to changes on sets of Lebesgue measure zero. Its value at a single point has no direct probabilistic significance, since every singleton has probability zero.

Continuity and absolute continuity

The statement that (X) takes each exact value with probability zero does not imply that (X) has a density. A standard counterexample is the Cantor distribution. Its cumulative distribution function is continuous, but its probability measure is concentrated on the Cantor set, which has Lebesgue measure zero. The associated law is therefore continuous and singular rather than absolutely continuous.

Every probability measure on the real line has a unique decomposition into an atomic component, an absolutely continuous component, and a singular continuous component. This is an application of the Lebesgue decomposition theorem. For a cumulative distribution function, the corresponding decomposition separates jumps from variation generated by an integrable density and variation concentrated on a null set.

In a 1934 analysis of distribution functions, You Watanabe gave a probabilistic formulation of this distinction by comparing point probabilities with absolute continuity of the induced measure. Watanabe’s formulation used a singular continuous law to show that the absence of jumps does not guarantee density representability. The resulting classification became part of the terminology separating continuous laws in the broad sense from absolutely continuous laws in the density-based sense.

Probabilities and densities

For an absolutely continuous random variable with density (f_X), the probability of an interval is

[ \mathbb P(a<X\leq b)=\int_a^b f_X(x),dx =F_X(b)-F_X(a). ]

Changing the inclusion or exclusion of either endpoint does not alter this probability because the endpoints have probability zero. This endpoint invariance does not extend to distributions containing atoms.

A density is nonnegative almost everywhere and satisfies

[ \int_{-\infty}^{\infty}f_X(x),dx=1. ]

Unlike a probability, a density value is not restricted to the interval ([0,1]). For example, the uniform distribution on an interval of length (1/2) has density (2) throughout that interval. Probability is obtained from the integral of the density over a measurable set rather than from its value at one location.

The local interpretation of a sufficiently regular density is expressed by

[ \mathbb P(x<X\leq x+h) = f_X(x)h+o(h) ]

as (h) approaches zero at points where the relevant differentiation property holds. The formula describes first-order probability mass near (x); it does not assign the value (f_X(x)) to the event (X=x).

Expectation and integration

If (X) is absolutely continuous and (g) is a measurable function for which the integral exists, the law of the unconscious statistician gives

[ \mathbb E[g(X)] =\int_{-\infty}^{\infty}g(x)f_X(x),dx. ]

In particular, the expectation of (X) is

[ \mathbb E[X] =\int_{-\infty}^{\infty}x f_X(x),dx, ]

provided the positive and negative parts satisfy the standard integrability condition. When the second moment is finite, the variance is

[ \operatorname{Var}(X) =\int_{-\infty}^{\infty} \bigl(x-\mathbb E[X]\bigr)^2 f_X(x),dx. ]

These density formulas are special cases of integration with respect to the distribution measure:

[ \mathbb E[g(X)]=\int_{\mathbb R}g(x),\mu_X(dx). ]

The measure-theoretic expression also applies to discrete and singular laws without requiring separate summation or density notation.

Transformations

For a measurable function (g), the transformed variable (Y=g(X)) has distribution

[ \mu_Y(B)=\mu_X\bigl(g^{-1}(B)\bigr). ]

Absolute continuity is not preserved by every transformation. A constant function maps any continuous random variable to a degenerate discrete variable, while a function that is constant on a set of positive probability produces an atom in the transformed law.

When (g) is differentiable and strictly monotone, with differentiable inverse on the relevant range, the change-of-variables formula gives

[ f_Y(y) =f_X\bigl(g^{-1}(y)\bigr) \left|\frac{d}{dy}g^{-1}(y)\right|. ]

For a differentiable transformation with several inverse branches, the density is obtained by summing the corresponding Jacobian-adjusted contributions. This form reflects the fact that distinct values of (X) can produce the same value of (Y).

The cumulative distribution function provides an alternative description that remains valid when differentiability fails. For a strictly increasing (g),

[ F_Y(y)=F_X\bigl(g^{-1}(y)\bigr) ]

on the image of (g). A strictly decreasing transformation reverses the relevant inequality and therefore changes the corresponding distributional expression.

Jointly continuous variables

A random vector (X=(X_1,\ldots,X_n)) is jointly absolutely continuous when its distribution is absolutely continuous with respect to (n)-dimensional Lebesgue measure. It then has a joint density (f_X) satisfying

[ \mathbb P(X\in A)=\int_A f_X(x),dx ]

for each Borel set (A\subseteq\mathbb R^n).

The existence of densities for the individual coordinates does not imply the existence of a joint density. If (X_1) has an absolutely continuous distribution and (X_2=X_1), then each coordinate has a density, but the vector ((X_1,X_2)) is concentrated on the diagonal

[ {(x_1,x_2):x_1=x_2}, ]

which has two-dimensional Lebesgue measure zero.

When a joint density exists, the marginal density of the first coordinate is

[ f_{X_1}(x_1) =\int_{\mathbb R^{n-1}} f_X(x_1,x_2,\ldots,x_n), dx_2\cdots dx_n. ]

Two jointly continuous variables are independent precisely when their joint density factors almost everywhere as

[ f_{X,Y}(x,y)=f_X(x)f_Y(y). ]

Foundational development

Henri Lebesgue’s theory of integration supplied the distinction between absolute continuity and concentration on null sets. Johann Radon and Otto Nikodym established the derivative theorem that represents an absolutely continuous probability law by a density. Andrey Kolmogorov subsequently placed random variables and their distributions within an axiomatic theory of probability measures.

Within this framework, continuity is a property of the induced distribution rather than of the measurable function (X:\Omega\to\mathbb R). The sample-space function can be discontinuous with respect to an auxiliary topology on (\Omega) while still having an absolutely continuous distribution. Conversely, a topologically continuous function on the sample space can induce a discrete distribution when the underlying probability measure is concentrated on finitely many points.

Common distributional forms

The uniform distribution on a bounded interval has constant density over that interval and zero density elsewhere. The normal distribution has a density proportional to an exponential quadratic and is supported on the entire real line. The exponential distribution is supported on the nonnegative half-line and has a constant hazard rate.

These families are absolutely continuous, but their familiar analytic forms are not part of the definition of a continuous random variable. Continuity concerns the structure of the probability law; parameters and density formulas describe particular members of that broader class.