MultinormalDistribution
✖
MultinormalDistribution
represents a multivariate normal distribution with zero mean and covariance matrix Σ.
represents a multivariate normal (Gaussian) distribution with mean vector μ and covariance matrix Σ.
Details

- The probability density for vector
in a multivariate normal distribution is proportional to
.
- MultinormalDistribution allows μ to be any vector of real numbers, and Σ any symmetric positive definite
×
matrix of real numbers with p=Length[μ].
- The mean vector μ and covariance matrix Σ can be quantities such that μ⊗μ and Σ have the same unit dimensions componentwise. »
- MultinormalDistribution can be used with such functions as Mean, CDF, and RandomVariate.
Background & Context
- MultinormalDistribution[μ,Σ] represents a continuous multivariate statistical distribution supported over the set of
of all
-tuples
and characterized by the property that each of the
(univariate) marginal distributions is a NormalDistribution for
. In other words, each of the variables
satisfies xkNormalDistribution for
. The multinormal distribution MultinormalDistribution[μ,Σ] is parametrized by a vector μ of real numbers and by a positive definite symmetric matrix Σ, which satisfy nLength[μ]Length[Σ] and which define the associated mean, variance, and covariance of the distribution. The multinormal distribution is sometimes referred to as the multivariate normal distribution, as a result of the fact that its univariate marginals are normally distributed.
- The probability density function (PDF) of a multinormal distribution has a single absolute maximum, though like the binormal distribution (BinormalDistribution) it may have multiple "peaks" (i.e. relative maxima). In general, the tails of each of the associated marginal PDFs are "thin" in the sense that the marginal PDF decreases exponentially rather than algebraically for large values of
. (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of these marginal distributions.)
- Most of the applications of the multinormal distribution correspond to the cases
(BinormalDistribution) or
rather than the general multinormal case. However, as a result of the multivariate central limit theorem, the multivariate normal distribution can be used to describe (at least qualitatively) any set of real-valued random variables, each of whose variates cluster around a given mean value. Even so, the bulk of the earliest literature (some of which dates back to the early 1800s) on multivariable extensions of the normal distribution focus on the bivariate and trivariate cases, which are applied in a wide range of fields, including genetics, materials science, evolutionary biology, economics, ecology, and medicine.
- RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a multinormal distribution. Distributed[x,MultinormalDistribution[μ,Σ]] , written more concisely as xMultinormalDistribution[μ,Σ], can be used to assert that a random variable x is distributed according to a multinormal distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
- The probability density and cumulative distribution functions for multinormal distributions may be given using PDF[MultinormalDistribution[μ,Σ],x] and CDF[MultinormalDistribution[μ,Σ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
- DistributionFitTest can be used to test if a given dataset is consistent with a multinormal distribution, EstimatedDistribution to estimate a multinormal parametric distribution from given data, and FindDistributionParameters to fit data to a multinormal distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic multinormal distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic multinormal distribution.
- TransformedDistribution can be used to represent a transformed multinormal distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a multinormal distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving multinormal distributions.
- MultinormalDistribution is related to a number of other distributions, including the NormalDistribution and BinormalDistribution, as discussed above. The one-dimensional marginals of a MultinormalDistribution have a NormalDistribution, while each of the multivariate marginals is again an instance of MultinormalDistribution. MultinormalDistribution is a limiting case of MultivariateTDistribution in the sense that PDF[MultinormalDistribution[{μ1,μ2},{{1,ρ},{ ρ,1}}],{x,y}] is precisely the limit of PDF[MultivariateTDistribution[{μ1,μ2},{{1,ρ},{ ρ,1}},ν],{x,y}] as ν∞. Moreover, MultinormalDistribution can be obtained from LogMultinormalDistribution by a transformation (TransformedDistribution). MultinormalDistribution is also related to RiceDistribution and, because of its relation to the univariate NormalDistribution, is also related to LogNormalDistribution, DavisDistribution, LogLogisticDistribution, ExponentialDistribution, WeibullDistribution, GompertzMakehamDistribution, ExtremeValueDistribution, and GammaDistribution.
Examples
open allclose allBasic Examples (4)Summary of the most common use cases

https://wolfram.com/xid/0b7elrv0641id7p2-oy2ojl


https://wolfram.com/xid/0b7elrv0641id7p2-689i2

Cumulative distribution function:

https://wolfram.com/xid/0b7elrv0641id7p2-7a4qq4


https://wolfram.com/xid/0b7elrv0641id7p2-fnm69u

https://wolfram.com/xid/0b7elrv0641id7p2-zza0nn


https://wolfram.com/xid/0b7elrv0641id7p2-1jvtw3


https://wolfram.com/xid/0b7elrv0641id7p2-sgrrls

https://wolfram.com/xid/0b7elrv0641id7p2-v9opdi

Scope (8)Survey of the scope of standard use cases
Generate a sample of pseudorandom vectors from a bivariate normal distribution:

https://wolfram.com/xid/0b7elrv0641id7p2-ljmvtp
Visualize the sample using a histogram:

https://wolfram.com/xid/0b7elrv0641id7p2-okqqe1

Distribution parameters estimation:

https://wolfram.com/xid/0b7elrv0641id7p2-45b7g2
Estimate the distribution parameters from sample data:

https://wolfram.com/xid/0b7elrv0641id7p2-zwyjci


https://wolfram.com/xid/0b7elrv0641id7p2-k1bc33

Skewness and kurtosis are constant vectors:

https://wolfram.com/xid/0b7elrv0641id7p2-l46snq


https://wolfram.com/xid/0b7elrv0641id7p2-14mzf9


https://wolfram.com/xid/0b7elrv0641id7p2-tsf0ki

https://wolfram.com/xid/0b7elrv0641id7p2-8yxe0v

https://wolfram.com/xid/0b7elrv0641id7p2-bxjlwn

The ImplicitRegion for 3D correlation coefficients of a multinormal distribution:

https://wolfram.com/xid/0b7elrv0641id7p2-uwfe5m

https://wolfram.com/xid/0b7elrv0641id7p2-lj6mc3

https://wolfram.com/xid/0b7elrv0641id7p2-cxciev
Use RandomPoint to sample from uniform distribution over 3D correlation coefficients region:

https://wolfram.com/xid/0b7elrv0641id7p2-b7b6ro

https://wolfram.com/xid/0b7elrv0641id7p2-dq3x

Estimate probability that determinant of the random correlation matrix is less than 0.1:

https://wolfram.com/xid/0b7elrv0641id7p2-bkgrm2


https://wolfram.com/xid/0b7elrv0641id7p2-y9qnwj

Univariate marginals follow a NormalDistribution:

https://wolfram.com/xid/0b7elrv0641id7p2-48ghft

https://wolfram.com/xid/0b7elrv0641id7p2-tnj6xf

Multivariate marginals follow a multivariate normal distribution:

https://wolfram.com/xid/0b7elrv0641id7p2-29y7hj

Consistent use of Quantity in parameters yields QuantityDistribution:

https://wolfram.com/xid/0b7elrv0641id7p2-ukzz1

Find standard deviations of each measurement:

https://wolfram.com/xid/0b7elrv0641id7p2-fxtz7g

Generalizations & Extensions (1)Generalized and extended use cases
MultinormalDistribution[Σ] is understood to have zero mean:

https://wolfram.com/xid/0b7elrv0641id7p2-etepg

Applications (2)Sample problems that can be solved with this function
Show a distribution function and its histogram in the same plot:

https://wolfram.com/xid/0b7elrv0641id7p2-ig1qzy
Compare the PDF to its histogram version:

https://wolfram.com/xid/0b7elrv0641id7p2-h0i29c

https://wolfram.com/xid/0b7elrv0641id7p2-ba9s1s

Compare the CDF to its histogram version:

https://wolfram.com/xid/0b7elrv0641id7p2-5806k

Specify multinormal distribution for values of three incoming currents to a 4-way node:

https://wolfram.com/xid/0b7elrv0641id7p2-py26pm

Find the distribution of the current leaving the node:

https://wolfram.com/xid/0b7elrv0641id7p2-gdarv3

Find the 95% confidence interval for the current value:

https://wolfram.com/xid/0b7elrv0641id7p2-daqey

Properties & Relations (10)Properties of the function, and connections to other functions
Equal probability contours for a bivariate normal distribution:

https://wolfram.com/xid/0b7elrv0641id7p2-o9k

The multinormal distribution is closed under affine transformation:

https://wolfram.com/xid/0b7elrv0641id7p2-uw25g7

https://wolfram.com/xid/0b7elrv0641id7p2-8t3p2g

https://wolfram.com/xid/0b7elrv0641id7p2-h7nhkv


https://wolfram.com/xid/0b7elrv0641id7p2-172rna

https://wolfram.com/xid/0b7elrv0641id7p2-32mxeg

Relationships to other distributions:

NormalDistribution is the univariate case of multinormal distribution:

https://wolfram.com/xid/0b7elrv0641id7p2-t7mes8


https://wolfram.com/xid/0b7elrv0641id7p2-o74ppn


https://wolfram.com/xid/0b7elrv0641id7p2-qdv2u2

BinormalDistribution is the two-dimensional case of multinormal distribution:

https://wolfram.com/xid/0b7elrv0641id7p2-l9jw29


https://wolfram.com/xid/0b7elrv0641id7p2-qbcd3p


https://wolfram.com/xid/0b7elrv0641id7p2-ssuls3

Multinormal distribution is the limit of MultivariateTDistribution as goes to
:

https://wolfram.com/xid/0b7elrv0641id7p2-pdcuvc


https://wolfram.com/xid/0b7elrv0641id7p2-43l1h3


https://wolfram.com/xid/0b7elrv0641id7p2-ztee3d

Multinormal distribution is related to RiceDistribution:

https://wolfram.com/xid/0b7elrv0641id7p2-0dy6kk

https://wolfram.com/xid/0b7elrv0641id7p2-60bpyx

LogMultinormalDistribution is a transformation of MultinormalDistribution:

https://wolfram.com/xid/0b7elrv0641id7p2-gov8at

https://wolfram.com/xid/0b7elrv0641id7p2-plabfy

NormalDistribution can be obtained from MultinormalDistribution:

https://wolfram.com/xid/0b7elrv0641id7p2-e76tlq

MultinormalDistribution is equivalent to CopulaDistribution with multinormal kernel and Gaussian marginals:

https://wolfram.com/xid/0b7elrv0641id7p2-pyufss
Corresponding multinormal distribution:

https://wolfram.com/xid/0b7elrv0641id7p2-gvyotj
The probability density functions are equal:

https://wolfram.com/xid/0b7elrv0641id7p2-hap7nq

Possible Issues (2)Common pitfalls and unexpected behavior
MultinormalDistribution is not defined when μ is not a vector of real numbers:

https://wolfram.com/xid/0b7elrv0641id7p2-bu4jit


MultinormalDistribution is not defined when the dimensions of μ and Σ are not consistent:

https://wolfram.com/xid/0b7elrv0641id7p2-dqrjwq


MultinormalDistribution is not defined when Σ is not symmetric and positive definite:

https://wolfram.com/xid/0b7elrv0641id7p2-ef2


Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:

https://wolfram.com/xid/0b7elrv0641id7p2-cbt

Wolfram Research (2010), MultinormalDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/MultinormalDistribution.html (updated 2016).
Text
Wolfram Research (2010), MultinormalDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/MultinormalDistribution.html (updated 2016).
Wolfram Research (2010), MultinormalDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/MultinormalDistribution.html (updated 2016).
CMS
Wolfram Language. 2010. "MultinormalDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/MultinormalDistribution.html.
Wolfram Language. 2010. "MultinormalDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/MultinormalDistribution.html.
APA
Wolfram Language. (2010). MultinormalDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MultinormalDistribution.html
Wolfram Language. (2010). MultinormalDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MultinormalDistribution.html
BibTeX
@misc{reference.wolfram_2025_multinormaldistribution, author="Wolfram Research", title="{MultinormalDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/MultinormalDistribution.html}", note=[Accessed: 08-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_multinormaldistribution, organization={Wolfram Research}, title={MultinormalDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/MultinormalDistribution.html}, note=[Accessed: 08-July-2025
]}