Multivariate normal cumulative distribution function
specifies control parameters for the numerical integration used to compute
p = mvncdf(___,
p, using any of the input argument combinations in the
previous syntaxes. Create the
options argument using the
statset function with any
combination of the parameters
Standard Multivariate Normal Distribution cdf
Evaluate the cdf of a standard four-dimensional multivariate normal distribution at points with increasing coordinates in every dimension.
Create a matrix
X of five four-dimensional points with increasing coordinates.
firstDim = (-2:2)'; X = repmat(firstDim,1,4)
X = 5×4 -2 -2 -2 -2 -1 -1 -1 -1 0 0 0 0 1 1 1 1 2 2 2 2
Evaluate the cdf at the points in
p = mvncdf(X)
p = 5×1 0.0000 0.0006 0.0625 0.5011 0.9121
The cdf values increase because the coordinates of the points are increasing in every dimension.
Bivariate Normal Distribution cdf
Compute and plot the cdf of a bivariate normal distribution.
Define the mean vector
mu and the covariance matrix
mu = [1 -1]; Sigma = [.9 .4; .4 .3];
Create a grid of 625 evenly spaced points in two-dimensional space.
[X1,X2] = meshgrid(linspace(-1,3,25)',linspace(-3,1,25)'); X = [X1(:) X2(:)];
Evaluate the cdf of the normal distribution at the grid points.
p = mvncdf(X,mu,Sigma);
Plot the cdf values.
Z = reshape(p,25,25); surf(X1,X2,Z)
Probability over Rectangular Region
Compute the probability over the unit square of a bivariate normal distribution, and create a contour plot of the results.
Define the bivariate normal distribution parameters
mu = [0 0]; Sigma = [0.25 0.3; 0.3 1];
Compute the probability over the unit square.
p = mvncdf([0 0],[1 1],mu,Sigma)
p = 0.2097
To visualize the result, first create a grid of evenly spaced points in two-dimensional space.
x1 = -3:.2:3; x2 = -3:.2:3; [X1,X2] = meshgrid(x1,x2); X = [X1(:) X2(:)];
Then, evaluate the pdf of the normal distribution at the grid points.
y = mvnpdf(X,mu,Sigma); y = reshape(y,length(x2),length(x1));
Finally, create a contour plot of the multivariate normal distribution that includes the unit square.
contour(x1,x2,y,[0.0001 0.001 0.01 0.05 0.15 0.25 0.35]) xlabel('x') ylabel('y') line([0 0 1 1 0],[1 0 0 1 1],'Linestyle','--','Color','k')
Computing a multivariate cumulative probability requires significantly more work than computing a univariate probability. By default, the
mvncdf function computes values to less than full machine precision, and returns an estimate of the error as an optional second output. View the error estimate in this case.
[p,err] = mvncdf([0 0],[1 1],mu,Sigma)
p = 0.2097
err = 1.0000e-08
Error Estimates in cdf Calculation
Evaluate the cdf of a multivariate normal distribution at random points, and display the error estimates associated with the cdf calculation.
Generate four random points from a five-dimensional multivariate normal distribution with mean vector
mu and covariance matrix
mu = [0.5 -0.3 0.2 0.1 -0.4]; Sigma = 0.5*eye(5); rng('default') % For reproducibility X = mvnrnd(mu,Sigma,4);
Find the cdf values
p at the points in
X and the associated error estimates
err. Display a summary of the numerical calculations.
[p,err] = mvncdf(X,mu,Sigma,statset('Display','final'))
Successfully satisfied error tolerance of 0.0001 in 8650 function evaluations. Successfully satisfied error tolerance of 0.0001 in 8650 function evaluations. Successfully satisfied error tolerance of 0.0001 in 8650 function evaluations. Successfully satisfied error tolerance of 0.0001 in 8650 function evaluations.
p = 4×1 0.1520 0.0407 0.0002 0.1970
err = 4×1 10-16 × 0.5949 0.1487 0 0.1983
X — Evaluation points
Evaluation points, specified as an
n-by-d numeric matrix, where
n is a positive scalar integer and
d is the dimension of a single multivariate normal
distribution. The rows of
X correspond to observations
(or points), and the columns correspond to variables (or
mu — Mean vector of multivariate normal distribution
vector of zeros (default) | numeric vector | numeric scalar
Mean vector of a multivariate normal distribution, specified as a
1-by-d numeric vector or a numeric
scalar, where d is the dimension of the multivariate
normal distribution. If
mu is a scalar, then
mvncdf replicates the scalar to match the size of
Sigma — Covariance matrix of multivariate normal distribution
identity matrix (default) | symmetric, positive definite matrix | numeric vector of diagonal entries
Covariance matrix of a multivariate normal distribution, specified as a
d-by-d symmetric, positive
definite matrix, where d is the dimension of the
multivariate normal distribution. If the covariance matrix is diagonal,
containing variances along the diagonal and zero covariances off it, then
you can also specify
Sigma as a
1-by-d vector containing just the
xl — Rectangle lower limit
Rectangle lower limit, specified as a
1-by-d numeric vector.
xu — Rectangle upper limit
Rectangle upper limit, specified as a
1-by-d numeric vector.
options — Numerical integration options
Numerical integration options, specified as a structure. Create the
options argument by calling the
statset function with any combination of the
'TolFun'— Maximum absolute error tolerance. The default value is
1e-8when d < 4, and
1e-4when d ≥ 4.
'MaxFunEvals'— Maximum number of integrand evaluations allowed when d ≥ 4. The default value is
1e7. The function ignores
'MaxFunEvals'when d < 4.
'Display'— Level of display output. The choices are
'final'. The function ignores
'Display'when d < 4.
err — Absolute error tolerance
positive numeric scalar
Absolute error tolerance, returned as a positive numeric scalar. For
bivariate and trivariate distributions, the default absolute error tolerance
1e-8. For four or more dimensions, the default
absolute error tolerance is
1e-4. For more information,
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
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
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.
In the one-dimensional case,
Sigmais the variance, not the standard deviation. For example,
mvncdf(1,0,4)is the same as
4is the variance and
2is the standard deviation.
For bivariate and trivariate distributions,
mvncdf uses adaptive
quadrature on a transformation of the t density, based on methods
developed by Drezner and Wesolowsky 
and by Genz . For four or more dimensions,
mvncdf uses a quasi-Monte Carlo integration algorithm based on
methods developed by Genz and Bretz 
 Drezner, Z. “Computation of the Trivariate Normal Integral.” Mathematics of Computation. Vol. 63, 1994, pp. 289–294.
 Drezner, Z., and G. O. Wesolowsky. “On the Computation of the Bivariate Normal Integral.” Journal of Statistical Computation and Simulation. Vol. 35, 1989, pp. 101–107.
 Genz, A. “Numerical Computation of Rectangular Bivariate and Trivariate Normal and t Probabilities.” Statistics and Computing. Vol. 14, No. 3, 2004, pp. 251–260.
 Genz, A., and F. Bretz. “Numerical Computation of Multivariate t Probabilities with Application to Power Calculation of Multiple Contrasts.” Journal of Statistical Computation and Simulation. Vol. 63, 1999, pp. 361–378.
 Genz, A., and F. Bretz. “Comparison of Methods for the Computation of Multivariate t Probabilities.” Journal of Computational and Graphical Statistics. Vol. 11, No. 4, 2002, pp. 950–971.
 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.
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).