mvnpdf: Multivariate Normal - Probablity Density Function

In jeksterslabds/jeksterslabRdist: Jekster's Lab Distribution Utilities

Description

Calculates probablities from the probability density function of the multivariate normal distribution \mathbf{X} \sim \mathcal{N}_{k} ≤ft( \boldsymbol{μ}, \boldsymbol{Σ} \right) . %(\#eq:dist-X-mvn)

Usage

mvnpdf(X, mu, Sigma)

Arguments

X	Numeric matrix. Values of the k-dimensional random variable \mathbf{X}.
mu	Numeric vector. Location parameter mean vector \boldsymbol{μ} of length k.
Sigma	Numeric matrix. k \times k variance-covariance matrix \boldsymbol{Σ}.

Details

The multivariate normal (or multivariate Gaussian, or joint normal) distribution is given by

\mathbf{X} \sim \mathcal{N}_{k} ≤ft( \boldsymbol{μ}, \boldsymbol{Σ} \right) %(\#eq:dist-X-mvn)

and has the probability density function (PDF)

f_{\mathbf{X}} ≤ft( X_1, …, X_k \right) = \frac{ \exp ≤ft[ - \frac{1}{2} ≤ft( \mathbf{X} - \boldsymbol{μ} \right)^{\prime} \boldsymbol{Σ}^{-1} ≤ft( \mathbf{X} - \boldsymbol{μ} \right) \right] }{ √{ ≤ft( 2 π \right)^{k} | \boldsymbol{Σ} | } } %(\#eq:dist-mvnpdf)

with k-dimensional random vector \mathbf{X} \in \boldsymbol{μ} + \textrm{span}≤ft(\boldsymbol{Σ}\right) \subseteq \mathbf{R}^k, \boldsymbol{μ} is the location parameter mean ≤ft( \boldsymbol{μ} \in \mathbf{R}^k \right), and \boldsymbol{Σ} is the variance-covariance matrix ≤ft( \boldsymbol{Σ} \in \mathbf{R}^{k \times k} \right).

Author(s)

Ivan Jacob Agaloos Pesigan

References

Wikipedia: Multivariate Normal Distribution

See Also

Other mutivariate normal likelihood functions: mvn2ll(), mvnll()

jeksterslabds/jeksterslabRdist documentation built on Aug. 9, 2020, 7:33 a.m.