Multivariate normal cumulative distribution function

returns the cumulative distribution function (cdf) of the multivariate normal
distribution with zero mean and identity covariance matrix, evaluated at each
row of `p`

= mvncdf(`X`

)`X`

. For more information, see Multivariate Normal Distribution.

specifies control parameters for the numerical integration used to compute
`p`

= mvncdf(___,`options`

)`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 `'TolFun'`

,
`'MaxFunEvals'`

, and `'Display'`

.

`[`

additionally returns an estimate of the error in `p`

,`err`

] = mvncdf(___)`p`

. For
more information, see Algorithms.

In the one-dimensional case,

`sigma`

is the variance, not the standard deviation. For example,`mvncdf(1,0,4)`

is the same as`normcdf(1,0,2)`

, where`4`

is the variance and`2`

is 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 [1]
[2]
and by Genz [3]. For four or more dimensions,
`mvncdf`

uses a quasi-Monte Carlo integration algorithm based on
methods developed by Genz and Bretz [4]
[5].

[1] Drezner, Z. “Computation of the Trivariate Normal
Integral.” *Mathematics of Computation*. Vol. 63, 1994, pp.
289–294.

[2] 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.

[3] Genz, A. “Numerical Computation of Rectangular
Bivariate and Trivariate Normal and t Probabilities.” *Statistics and
Computing*. Vol. 14, No. 3, 2004, pp. 251–260.

[4] 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.

[5] 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.

[6] 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.