Mathematica noob here, and I'm having some issues plotting 90% confidence ellipses of a small-variance 2D Gaussian PDF. I have it working perfectly fine for larger covariance matrices, but absolutely no luck with these tiny ones! Here's my code:

Sigma={{0.0093041, -0.00126552}, {-0.00126552, 0.00020695}}; xv={X,Y}; x0={66.2782, 0.791879}; pdf = 1/Sqrt[(2 \[Pi])^Length[xv] Det[Sigma]]Exp[-(xv - x0).Inverse[Sigma].(xv - x0)/2] ContourPlot[pdf, {X, 66, 66.6}, {Y, .76, .83}, PlotRange -> All, PlotPoints -> 100] 

Which gives me a standard multivariate Gaussian, albiet with contours at probabilities of 50-250.. Which I guess is fine assuming such a small range. Also, just as a quick check, the integral does indeed equal 1:

NIntegrate[pdf, {X, 60, 70}, {Y, .5, .9}](*=1*) 

Now, trying to compute the 90% confidence regions.. Following the solution here: https://mathematica.stackexchange.com/a/20481/58658, I do the following:

f[x_, y_, t_: 0] := With[{z = pdf /. X -> x /. Y -> y}, z Boole[z >= t]]; r = FindRoot[NIntegrate[f[x, y, t], {x, 60, 70}, {y, .7, .9}, AccuracyGoal -> 3, PrecisionGoal -> 6] - 0.90, {t, 0, 2}, Method -> "Brent", AccuracyGoal -> 3, PrecisionGoal -> 6]; 

Which results in numerous errors "The integrand has evaluated to non-numerical values for all sampling points in the region..", or non-sensical answers as I play around with the integration ranges. I don't really see how to make this work, seeing as all the pdf values evaluate to be much greater than 1 in such a small range!! Does anyone have any experience with issues like this? Or any clever roundabout ways of doing this?

Thanks so much in advance!

Mathematica noob here, and I'm having some issues plotting 90% confidence ellipses of a small-variance 2D Gaussian PDF. I have it working perfectly fine for larger covariance matrices, but absolutely no luck with these tiny ones! Here's my code:

Sigma={{0.0093041, -0.00126552}, {-0.00126552, 0.00020695}}; xv={X,Y}; x0={66.2782, 0.791879}; pdf = 1/Sqrt[(2 \[Pi])^Length[xv] Det[Sigma]]Exp[-(xv - x0).Inverse[Sigma].(xv - x0)/2] ContourPlot[pdf, {X, 66, 66.6}, {Y, .76, .83}, PlotRange -> All, PlotPoints -> 100] 

Which gives me a standard multivariate Gaussian, albiet with contours at densities of 50-250. Which I guess is fine assuming such a small range. Also, just as a quick check, the integral does indeed equal 1:

NIntegrate[pdf, {X, 60, 70}, {Y, .5, .9}](*=1*) 

Now, trying to compute the 90% confidence regions.. Following the solution here: https://mathematica.stackexchange.com/a/20481/58658, I do the following:

f[x_, y_, t_: 0] := With[{z = pdf /. X -> x /. Y -> y}, z Boole[z >= t]]; r = FindRoot[NIntegrate[f[x, y, t], {x, 60, 70}, {y, .7, .9}, AccuracyGoal -> 3, PrecisionGoal -> 6] - 0.90, {t, 0, 2}, Method -> "Brent", AccuracyGoal -> 3, PrecisionGoal -> 6]; 

Which results in numerous errors "The integrand has evaluated to non-numerical values for all sampling points in the region..", or non-sensical answers as I play around with the integration ranges. I don't really see how to make this work, seeing as all the pdf densities evaluate to be much greater than 1 in such a small range!! Does anyone have any experience with issues like this? Or any clever roundabout ways of doing this?

Thanks so much in advance!