I am fitting a bivariate skew normal distribution to a 2D data through the sn package in R. I get a $2 \times 1$ vector of location and shape parameters $\beta$ and $\alpha$ as well as a $2 \times 2$ scale matrix $\omega$.
In the case of univariate skew normal distribution, I understand the expectation and the variance is given by,
$$ \mu_x = \beta + \omega\delta \sqrt{ \frac{2}{\pi}} $$ and
$$ \sigma_x^2 = \omega^2\left(1-\frac{2\delta^2}{\pi}\right) $$
where
$$ \delta = \frac{\alpha}{\sqrt{1+\alpha^2}} $$
and I can compute these quantities when we get the parameters fitting a univariate random variable.
However, here we have bivariate distribution - I have vector quantities and I think we will end up with a vector of expectation and variance, one for each dimension. I am not sure how to compute them through the parameters I got through fitting. Specifically, how do I compute $\delta$ and $\delta^2$ when we have vectors. Please help.