joint normal distribution

A finite setMathworldPlanetmath of random variablesMathworldPlanetmath X1,,Xn are said to have a joint normal distribution or multivariate normal distribution if all real linear combinationsMathworldPlanetmath


are normal ( This implies, in particular, that the individual random variables Xi are each normally distributed. However, the converseMathworldPlanetmath is not not true and sets of normally distributed random variables need not, in general, be jointly normal.

If 𝑿=(X1,X2,,Xn) is joint normal, then its probability distribution is uniquely determined by the means 𝝁n and the n×n positive semidefinite covariance matrixMathworldPlanetmath 𝚺,


Then, the joint normal distribution is commonly denoted as N(𝝁,𝚺). Conversely, this distributionDlmfPlanetmathPlanetmath exists for any such 𝝁 and 𝚺.

Figure 1: DensityPlanetmathPlanetmath of joint normal variables X,Y with Var(X)=2, Var(Y)=1 and Cov(X,Y)=-1.

The joint normal distribution has the following properties:

  1. 1.

    If 𝑿 has the N(𝝁,𝚺) distribution for nonsigular 𝚺 then it has the multidimensional Gaussian probability density function

  2. 2.

    If 𝑿 has the N(𝝁,𝚺) distribution and 𝝀n then

  3. 3.

    Sets of linear combinations of joint normals are themselves joint normal. In particular, if 𝑿N(𝝁,𝚺) and A is an m×n matrix, then A𝑿 has the joint normal distribution N(A𝝁,A𝚺AT).

  4. 4.

    The characteristic functionMathworldPlanetmathPlanetmathPlanetmathPlanetmath is given by


    for 𝑿N(𝝁,𝚺) and any 𝒂n.

  5. 5.

    A pair X,Y of jointly normal random variables are independent if and only if they have zero covarianceMathworldPlanetmath.

  6. 6.

    Let 𝑿 be a random vector whose distribution is jointly normal. Suppose the coordinatesPlanetmathPlanetmath of 𝑿 are partitioned into two groups, forming random vectors 𝑿𝟏 and 𝑿𝟐, then the conditional distribution of 𝑿𝟏 given 𝑿𝟐=𝒄 is jointly normal.

Title joint normal distribution
Canonical name JointNormalDistribution
Date of creation 2013-03-22 15:22:34
Last modified on 2013-03-22 15:22:34
Owner gel (22282)
Last modified by gel (22282)
Numerical id 14
Author gel (22282)
Entry type Definition
Classification msc 62H05
Classification msc 60E05
Synonym multivariate Gaussian distribution
Related topic NormalRandomVariable
Defines jointly normal
Defines multivariate normal distribution