The following is Sheldon Ross's definition:
We say that the random variables $X,Y$ have a bivariate normal
distribution if, for some constants
$\mu_x,\mu_y,\sigma_x>0,\sigma_y>0, -1<\rho < 1$, their joint density
function is given, for all $-\infty < x,y < \infty$, by
$$f(x,y)=\frac{\exp\left(-\frac1{2(1-\rho^2)}\left(\left(\frac{x-\mu_x}{\sigma_x}\right)^2+\left(\frac{y-\mu_y}{\sigma_y}\right)^2-2\rho\frac{(x-\mu_x)(y-\mu_y)}{\sigma_x\sigma_y}\right)\right)}{2\pi\sigma_x\sigma_y\sqrt{1-\rho^2}}$$
Is there a combinatorial/intuitive meaning of this definition?
Best Answer
I don't have a combinatorial meaning, but you can think of it as follows. $(X,Y)$ is the result of applying an affine transformation to a pair $(W,Z)$ of independent standard normal random variables. Many such transformations exist, and one in particular is
$$\begin{align*} X &= \mu_x + \sigma_x W\\ Y &= \mu_y + \rho \sigma_y W + \sqrt{1-\rho^2} \sigma_y Z \end{align*}$$
See for example this set of slides. The contours of the joint density (points at equal height above the $x$-$y$ plane) are ellipses centered at $(\mu_x,\mu_y)$.