Confidence ellipse for a 2D gaussian

Question

For a 1D gaussian, the interval +/- 1SD about the mean will comprise ~68% of the area under the curve. Consider a 2D gaussian with a mean of zero and a diagonal covariance matrix (i.e., it is not rotated), with standard deviations $\sigma_x$ and $\sigma_y$. Consider the ellipse drawn to pass through the points $(\pm\sigma_x, 0)$ and $(0, \pm\sigma_y)$. Will it also contain 68% of the volume underneath the 2D curve? I found this question which asks a more general version of this question, but I couldn't immediately tell from the reference given what the answer was.

Answer 1

First observe that the required probability $<\ \approx 0.68^2=0.4624$

Then observe that you might as well assume $\sigma_1=\sigma_2=1$, then your question becomes what is the probability that a point lies inside the unit circle. Then proceed by converting to polars, when: $$ p=\int_{r=0}^1 f_R(r)\;dr=\int_{\theta=0}^{2\pi} \int_{r=0}^1\frac{1}{2\pi}e^{-r^2/2}\; rdrd\theta=1-\frac{1}{\sqrt{e}}\approx 0.393 $$