I'm taking a stat class and the professor just wrote down the joint pdf of a bivariate normal distribution, even though the pdf looks "intuitive" but I'm not able to derive the formula for the same.
I'm unsure of where I even start.
Here's what it looks like :
$$ f(x, y) = \frac{1}{2\pi\sigma_x\sigma_y\sqrt{1-\rho^2}} \exp\left(-\frac{1}{2(1-\rho^2)}\left[\frac{(x - \mu_x)^2}{\sigma_x^2} - 2\rho\frac{(x - \mu_x)(y - \mu_y)}{\sigma_x\sigma_y} + \frac{(y - \mu_y)^2}{\sigma_y^2}\right]\right) $$
Where do I start given I know :
$$ f(x) = \frac{1}{\sigma_x\sqrt{2\pi}} \exp\left(-\frac{(x - \mu_x)^2}{2\sigma_y^2}\right) $$
$$ f(y) = \frac{1}{\sigma_y\sqrt{2\pi}} \exp\left(-\frac{(y - \mu_y)^2}{2\sigma_y^2}\right) $$
and
$$\rho = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} $$