I am currently attempting to use a bivariate normal distribution to identify the most likely range of movement for a blob in computer vision. This itself is not the problem, however; I do not understand how σ plays a role in finding discrete probability contours.
I am not permitted to post images yet since my reputation is too low, but here is a link to a the graph:
This is a sample contour plot from Mathematica which displays the Bivariate Probability Density Function with σX = .27 and σY = .54 . μX = 0, μY = 0, and ρ = 0.
I would appreciate it very much if someone could explicate what determines the contour ellipses and how I would go about calculating them for functions of variable σX and σY.
I am not a Mathematica expert, but it seems as though the values for the level curves were selected such that the level curves represent five even steps from the peak at (0,0) to where the surface "levels out."
Regardless, let's look at the bivariate Gaussian distribution for $\rho = 0$, which implies that $X$ and $Y$ are uncorrelated.
You can write the PDF of this distribution as
$f(x,y) = \frac{1}{2\pi\sigma_x\sigma_y} \exp \left(-\frac{1}{2}\left[\frac{x-\mu_x}{\sigma_x^2}+\frac{y-\mu_y}{\sigma_y^2}-\frac{2(x-\mu_x)(y-\mu_y)}{\sigma_x \sigma_y}\right]\right).$
You can compute this surface in a straightforward manner, and use any contour curve generating algorithm to plot those curves.