Bivariate Skewed Normal Distribution What is the equation for a multivariate skewed normal distribution, specifically a two dimensional skewed normal distribution?
 A: 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
$$
    \phi_{\text{Skew}}(x;\alpha) =2\phi(x)\Phi(\alpha x)
$$
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")

