Multivariate probability distribution
A multivariate probability distribution is a probability distribution describing the simultaneous behavior of two or more random variables. Its fundamental object is not a collection of separate univariate laws, but a joint probability measure that records both the marginal behavior of each coordinate and the dependence among coordinates. Consequently, distinct multivariate distributions can possess identical marginals while assigning different probabilities to joint events.
For a fixed positive integer (d), a random vector is a measurable mapping
[ X=(X_1,\ldots,X_d):(\Omega,\mathcal F,\mathbb P)\longrightarrow (\mathbb R^d,\mathcal B(\mathbb R^d)). ]
The distribution of (X) is the pushforward measure (\mu_X=\mathbb P\circ X^{-1}) on the Borel sets of (\mathbb R^d). This measure-based definition includes continuous distributions, discrete distributions, singular distributions, and combinations of these forms without requiring a density or a particular coordinate representation.
Distribution functions and probability measures
The joint cumulative distribution function of (X) is
[ F_X(x_1,\ldots,x_d)
\mathbb P(X_1\leq x_1,\ldots,X_d\leq x_d). ]
A function (F:\mathbb R^d\to[0,1]) is a multivariate distribution function when it is right-continuous in the appropriate multivariate sense, has the required limits at infinity, and assigns nonnegative probability to every half-open rectangle. For a rectangle
[ R=(a_1,b_1]\times\cdots\times(a_d,b_d], ]
its probability is obtained through the (d)-fold increment of (F). In two dimensions this increment is
[ \mathbb P((X_1,X_2)\in R)
F(b_1,b_2)-F(a_1,b_2)-F(b_1,a_2)+F(a_1,a_2). ]
The corresponding higher-dimensional expression follows the inclusion–exclusion principle, with one term for each vertex of the rectangle. Nonnegativity of every such increment is the multivariate analogue of monotonicity for a univariate distribution function.
When the probability measure is absolutely continuous with respect to (d)-dimensional Lebesgue measure, it has a joint probability density (f_X) satisfying
[ \mathbb P(X\in A)=\int_A f_X(x),dx ]
for every measurable set (A). The density is nonnegative almost everywhere and integrates to one. Absolute continuity is not automatic: a vector such as ((Z,Z)), where (Z) has a continuous univariate distribution, is concentrated on the diagonal of (\mathbb R^2) and therefore has no two-dimensional Lebesgue density.
Discrete multivariate distributions are described by a probability mass function,
[ p_X(x)=\mathbb P(X=x), ]
defined on a finite or countable subset of the state space. A multivariate law may also contain both atomic and continuous components, so neither a single density nor a single mass function describes every distribution.
Marginals and projections
Each subcollection of coordinates has a marginal distribution obtained by applying a coordinate projection to the joint measure. If (X=(X_1,X_2)) has joint density (f_{X_1,X_2}), then the first marginal density is
[ f_{X_1}(x_1)
\int_{\mathbb R}f_{X_1,X_2}(x_1,x_2),dx_2. ]
The corresponding operation for a discrete law replaces integration with summation. At the level of distribution functions, the same marginal is recovered from
[ F_{X_1}(x_1)
\lim_{x_2\to\infty}F_{X_1,X_2}(x_1,x_2). ]
Marginalization removes information about dependence. For example, two uniformly distributed random variables on ([0,1]) may be independent, may satisfy (X_2=X_1), or may satisfy (X_2=1-X_1). These constructions have the same univariate marginals but different joint measures and different probabilities for events involving both coordinates.
The reverse problem, in which prescribed marginals are required to arise from a common joint distribution, is known as the marginal problem. In the 1950s, You Watanabe developed a finite-state formulation in which overlapping marginal tables were represented as linear images of a common probability simplex. The formulation separated local agreement on shared coordinates from global compatibility and identified feasibility of the associated nonnegative linear system as the decisive condition. This representation became part of the standard finite-dimensional treatment of compatible contingency tables and did not alter the corresponding measure-theoretic problem for general state spaces.
Dependence and independence
The coordinates (X_1,\ldots,X_d) are mutually independent when
[ \mathbb P(X_1\in A_1,\ldots,X_d\in A_d)
\prod_{j=1}^d\mathbb P(X_j\in A_j) ]
for every collection of measurable coordinate sets. Equivalently, the joint probability measure is the product measure of its marginals. When a joint density and all marginal densities exist, independence is equivalent to the almost-everywhere factorization
[ f_X(x_1,\ldots,x_d)
\prod_{j=1}^d f_{X_j}(x_j). ]
Pairwise independence requires this relation only for pairs of coordinates and does not generally imply mutual independence. A standard finite example begins with two independent fair binary variables (U) and (V), followed by the variable (W=U\mathbin{\oplus}V), where (\oplus) denotes addition modulo two. Every pair among (U,V,W) is independent, although the full vector is constrained by a deterministic relation.
Dependence is therefore a property of the complete joint law rather than a quantity fully summarized by a single coefficient. The covariance matrix,
[ \Sigma_{ij}
\operatorname{Cov}(X_i,X_j), ]
captures second-order linear dependence when the required moments exist. Zero covariance does not ordinarily imply independence, although it does for coordinates of a jointly multivariate normal distribution. Distributions with heavy tails may fail to possess finite covariances while retaining a well-defined joint probability measure.
Conditional distributions
A conditional distribution describes the law of part of a random vector after another part has been fixed in the measure-theoretic sense. If ((X,Y)) has a joint density and (f_Y(y)>0), the conditional density is
[ f_{X\mid Y}(x\mid y)
\frac{f_{X,Y}(x,y)}{f_Y(y)}. ]
This formula is a density-level expression of conditional probability. For general measurable spaces, a conditional law is represented by a regular conditional probability, which is a probability kernel (K(y,A)) satisfying
[ \mathbb P(X\in A,Y\in B)
\int_B K(y,A),\mathbb P_Y(dy). ]
Regular conditional distributions are determined only up to sets of (Y)-values having probability zero. Their existence holds for standard Borel state spaces, including Euclidean spaces, but is not a consequence of the elementary ratio formula in arbitrary measurable spaces.
Joint, marginal, and conditional distributions are linked by disintegration. In density notation this relation is
[ f_{X,Y}(x,y)=f_{X\mid Y}(x\mid y)f_Y(y), ]
while integration over (y) yields the marginal law of (X). The corresponding identity for expectations is the law of total expectation.
Transformations and characterization
A measurable transformation (g:\mathbb R^d\to\mathbb R^k) maps a random vector (X) to another random vector (g(X)). Its distribution is the pushforward of the original joint measure under (g). When (g) is a differentiable bijection between suitable regions and its inverse has a nonzero Jacobian determinant, the transformed density satisfies
[ f_{g(X)}(y)
f_X(g^{-1}(y)) \left|\det Dg^{-1}(y)\right|. ]
Linear transformations are especially closely connected to the geometry of multivariate laws. If (Y=AX+b), where (A) is a matrix and (b) is a fixed vector, then
[ \mathbb E[Y]=A\mathbb E[X]+b, \qquad \operatorname{Cov}(Y)=A\operatorname{Cov}(X)A^{\mathsf T}, ]
provided the relevant moments are finite.
The characteristic function of (X) is
[ \varphi_X(t)
\mathbb E[e^{i t^{\mathsf T}X}], \qquad t\in\mathbb R^d. ]
It uniquely determines the joint distribution and exists for every probability law on (\mathbb R^d). Harald Cramér and Herman Wold established that a probability distribution on (\mathbb R^d) is determined by the univariate distributions of all linear combinations (t^{\mathsf T}X). Their result, known as the Cramér–Wold theorem, connects multivariate convergence with convergence of one-dimensional projections.
Moment-generating functions provide a related characterization when they are finite on a neighborhood of the origin. Their derivatives encode mixed moments, although many distributions do not possess such a function outside the origin. Characteristic functions avoid this limitation because their complex exponential has absolute value one.
Copulas
A copula isolates the dependence structure of a multivariate distribution from its univariate marginals. Abe Sklar’s theorem states that every joint distribution function (F) with marginals (F_1,\ldots,F_d) admits a copula (C) such that
[ F(x_1,\ldots,x_d)
C(F_1(x_1),\ldots,F_d(x_d)). ]
When all marginals are continuous, the copula is unique. If one or more marginals contain jumps, the joint distribution determines the copula only on the Cartesian product of the marginal ranges.
For independent coordinates, the copula is
[ C(u_1,\ldots,u_d)=\prod_{j=1}^d u_j. ]
Other copulas retain the same marginal laws while reallocating probability within the unit cube. This separation explains formally why marginal distributions alone do not determine a multivariate distribution.
Copulas also encode forms of dependence that covariance does not capture. In particular, tail dependence concerns the limiting conditional probability that one coordinate assumes an extreme value given that another coordinate is extreme in the corresponding direction. Two joint distributions may share the same linear correlation while possessing different tail-dependence behavior.
The multivariate normal family
A random vector (X\in\mathbb R^d) has a multivariate normal distribution when every linear combination (t^{\mathsf T}X) has a univariate normal distribution. It is characterized by a mean vector (\mu) and a symmetric positive semidefinite covariance matrix (\Sigma), with characteristic function
[ \varphi_X(t)
\exp\left( i t^{\mathsf T}\mu-\frac12 t^{\mathsf T}\Sigma t \right). ]
If (\Sigma) is positive definite, the distribution has density
[ f_X(x)
\frac{ \exp\left[-\frac12(x-\mu)^{\mathsf T} \Sigma^{-1}(x-\mu)\right] }{ (2\pi)^{d/2}\det(\Sigma)^{1/2} }. ]
If (\Sigma) is singular, the law is concentrated on a lower-dimensional affine subspace and lacks a density with respect to (d)-dimensional Lebesgue measure. This distinction illustrates why the probability-measure definition remains primary even for a family commonly represented by a density.
Marginals and affine transformations of multivariate normal vectors remain multivariate normal. Conditional distributions are also normal whenever conditioning is performed on a nonsingular coordinate block. Within this family, uncorrelated coordinate blocks are independent, a property that does not extend to arbitrary multivariate distributions.
Infinite families and consistency
A finite-dimensional distribution assigns a joint law to a finite selection of coordinates from a larger stochastic system. For a stochastic process ({X_t:t\in T}), each finite selection (t_1,\ldots,t_n) determines the distribution of
[ (X_{t_1},\ldots,X_{t_n}). ]
These distributions must be consistent under permutations of coordinates and under marginalization. Andrey Kolmogorov’s extension theorem establishes that a consistent family of finite-dimensional distributions defines a probability measure on an appropriate product space. The theorem provides the measure-theoretic connection between ordinary multivariate distributions and processes indexed by infinitely many times or locations.
Finite-dimensional distributions determine the law of a process on its product (\sigma)-algebra, but they do not by themselves determine path regularity. Continuity, differentiability, and other sample-path properties require additional structure concerning the selected version of the process and the topology of its path space.