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Multivariate normal probability density function

`y = mvnpdf(X)`

y = mvnpdf(X,MU)

y = mvnpdf(X,MU,SIGMA)

`y = mvnpdf(X)`

returns the *n*-by-`1`

vector `y`

,
containing the probability density of the multivariate normal distribution
with zero mean and identity covariance matrix, evaluated at each row
of the *n*-by-*d* matrix `X`

.
Rows of `X`

correspond to observations and columns
correspond to variables or coordinates.

`y = mvnpdf(X,MU)`

returns
the density of the multivariate normal distribution with mean `mu`

and
identity covariance matrix, evaluated at each row of `X`

. `MU`

is
a `1`

-by-*d* vector, or an *n*-by-*d* matrix.
If `MU`

is a matrix, the density is evaluated for
each row of `X`

with the corresponding row of `MU`

. `MU`

can
also be a scalar value, which `mvnpdf`

replicates
to match the size of `X`

.

`y = mvnpdf(X,MU,SIGMA)`

returns
the density of the multivariate normal distribution with mean `MU`

and
covariance `SIGMA`

, evaluated at each row of `X`

. `SIGMA`

is
a *d*-by-*d* matrix, or a *d*-by-*d*-by-*n* array,
in which case the density is evaluated for each row of `X`

with
the corresponding page of `SIGMA`

, i.e., `mvnpdf`

computes `y(i)`

using `X(i,:)`

and `SIGMA(:,:,i)`

.
If the covariance matrix is diagonal, containing variances along the
diagonal and zero covariances off the diagonal, `SIGMA`

may
also be specified as a `1`

-by-*d* vector
or a 1-by-*d*-by-*n* array, containing
just the diagonal. Specify `[]`

for `MU`

to
use its default value when you want to specify only `SIGMA`

.

If `X`

is a 1-by-*d* vector, `mvnpdf`

replicates
it to match the leading dimension of `mu`

or the
trailing dimension of `SIGMA`

.

mu = [1 -1]; SIGMA = [.9 .4; .4 .3]; X = mvnrnd(mu,SIGMA,10); p = mvnpdf(X,mu,SIGMA);

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