Multivariate normal random numbers



R = mvnrnd(mu,Sigma,n) returns a matrix R of n random vectors chosen from the same multivariate normal distribution, with mean vector mu and covariance matrix Sigma. For more information, see Multivariate Normal Distribution.


R = mvnrnd(mu,Sigma) returns an m-by-d matrix R of random vectors sampled from m separate d-dimensional multivariate normal distributions, with means and covariances specified by mu and Sigma, respectively. Each row of R is a single multivariate normal random vector.


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Generate random numbers from the same multivariate normal distribution.

Define mu and Sigma, and generate 100 random numbers.

mu = [2 3];
Sigma = [1 1.5; 1.5 3];
rng('default')  % For reproducibility
R = mvnrnd(mu,Sigma,100);

Plot the random numbers.


Randomly sample from five different three-dimensional normal distributions.

Specify the means mu and the covariances Sigma of the distributions. Let all the distributions share the same covariance matrix, but vary the mean vectors.

firstDim = (1:5)';
mu = repmat(firstDim,1,3)
mu = 5×3

     1     1     1
     2     2     2
     3     3     3
     4     4     4
     5     5     5

Sigma = eye(3)
Sigma = 3×3

     1     0     0
     0     1     0
     0     0     1

Randomly sample once from each of the five distributions.

rng('default')  % For reproducibility
R = mvnrnd(mu,Sigma)
R = 5×3

    1.5377   -0.3077   -0.3499
    3.8339    1.5664    5.0349
    0.7412    3.3426    3.7254
    4.8622    7.5784    3.9369
    5.3188    7.7694    5.7147

Plot the results.


Input Arguments

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Means of multivariate normal distributions, specified as a 1-by-d numeric vector or an m-by-d numeric matrix.

  • If mu is a vector, then mvnrnd replicates the vector to match the trailing dimension of Sigma.

  • If mu is a matrix, then each row of mu is the mean vector of a single multivariate normal distribution.

Data Types: single | double

Covariances of multivariate normal distributions, specified as a d-by-d symmetric, positive semi-definite matrix or a d-by-d-by-m numeric array.

  • If Sigma is a matrix, then mvnrnd replicates the matrix to match the number of rows in mu.

  • If Sigma is an array, then each page of Sigma, Sigma(:,:,i), is the covariance matrix of a single multivariate normal distribution and, therefore, is a symmetric, positive semi-definite matrix.

If the covariance matrices are diagonal, containing variances along the diagonal and zero covariances off it, then you can also specify Sigma as a 1-by-d vector or a 1-by-d-by-m array containing just the diagonal entries.

Data Types: single | double

Number of multivariate random numbers, specified as a positive scalar integer. n specifies the number of rows in R.

Data Types: single | double

Output Arguments

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Multivariate normal random numbers, returned as one of the following:

  • m-by-d numeric matrix, where m and d are the dimensions specified by mu and Sigma

  • n-by-d numeric matrix, where n is the specified input argument and d is the dimension specified by mu and Sigma

If mu is a matrix and Sigma is an array, then mvnrnd computes R(i,:) using mu(i,:) and Sigma(:,:,i).

More About

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Multivariate Normal Distribution

The multivariate normal distribution is a generalization of the univariate normal distribution to two or more variables. It has two parameters, a mean vector μ and a covariance matrix Σ, that are analogous to the mean and variance parameters of a univariate normal distribution. The diagonal elements of Σ contain the variances for each variable, and the off-diagonal elements of Σ contain the covariances between variables.

The probability density function (pdf) of the d-dimensional multivariate normal distribution is

y = f(x,μ,Σ) = 1|Σ|(2π)dexp(12(x-μΣ-1(x-μ)')

where x and μ are 1-by-d vectors and Σ is a d-by-d symmetric, positive definite matrix. Only mvnrnd allows positive semi-definite Σ matrices, which can be singular. The pdf cannot have the same form when Σ is singular.

The multivariate normal cumulative distribution function (cdf) evaluated at x is the probability that a random vector v, distributed as multivariate normal, lies within the semi-infinite rectangle with upper limits defined by x:


Although the multivariate normal cdf does not have a closed form, mvncdf can compute cdf values numerically.


  • mvnrnd requires the matrix Sigma to be symmetric. If Sigma has only minor asymmetry, you can use (Sigma + Sigma')/2 instead to resolve the asymmetry.

  • In the one-dimensional case, Sigma is the variance, not the standard deviation. For example, mvnrnd(0,4) is the same as normrnd(0,2), where 4 is the variance and 2 is the standard deviation.


[1] Kotz, S., N. Balakrishnan, and N. L. Johnson. Continuous Multivariate Distributions: Volume 1: Models and Applications. 2nd ed. New York: John Wiley & Sons, Inc., 2000.

Extended Capabilities

Introduced before R2006a