# Solved – Bivariate Skewed Normal Distribution

What is the equation for a multivariate skewed normal distribution, specifically a two dimensional skewed normal distribution?

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Bivariate (or multivariate) skew normal distributions can be constructed with the same methods that is used in the univariate case. The usual univariate skewnormal density (due to Azzalini https://en.wikipedia.org/wiki/Skew_normal_distribution) is given by \$\$ phi_{text{Skew}}(x;alpha) =2phi(x)Phi(alpha x) \$\$ where \$phi\$ is the usual standard normal density and \$alpha\$ is a new skewness parameter. \$Phi\$ is the standard normal cumulative distribution.

We can use the same construction in the multivariate case, introducing the covariance matrix \$Omega\$ but still keeping the center at zero. \$\$ phi_{d,text{Skew}}(x;Omega,alpha) = 2 phi_d(x;Omega)Phi(alpha^T x) \$\$ where \$d\$ is the dimension and \$phi_d\$ is the multinormal density with covariance matrix \$Omega\$ (and center zero), \$Phi\$ is still the univariate standard normal cumulative distribution.

A contour plot is shown below, the parameters used can be gleaned from the R code below it: ``library(sn)    alpha <-  c(0.5, 1) Omega <-  matrix(c(1, 0.5, 0.5, 1), 2, 2) xran  <-  seq(-3, 3, length=101) yran  <-  seq(-3, 3, length=101) z     <-  outer(xran, yran, FUN=Vectorize( function(x, y) dmsn(c(x, y), c(0, 0),                                                     Omega, alpha) )  ) image(xran, yran, z) contour(xran, yran, z, ncontours=20, add=TRUE) title("bivariate skewnormal density") ``

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