I have both a 2D 'MultinormalDistribution', and also a single xy point, and I would like to be able to calculate the probability of this point (given the multinormal distribution) and also plot an ellipse with the same probability as this point around the mean of the distribution.
So far I have defined the Multinormal (xy) distribution like:
bvn = MultinormalDistribution[{-3.89764, 1.29137}, {{0.08369444426^2, 0}, {0, 0.0175124^2}}]
I also have the fixed point in this xy plane, at -3.98529, 1.30482. Previously (thanks to help from users ubpdqn and Karsten 7 in a previous question) I approached this by essentially doing two-tailed Gaussian integrals in both of the normal distributions that make up the multinormal distribution, which gave me the probability inside a square, represented visually with:
Plot3D[PDF[bvn, {x, y}], {x, -4.5, -3.5}, {y, 1.2, 1.4}, PlotRange -> All, MeshFunctions -> {#1 &, #2 &}, Mesh -> {{xl = -3.98529, xu = -3.80999}, {yl = 1.27792, yu = 1.30482}}]
I realize now that this square isn't exactly what I want, and instead I want to define an ellipse of constant probability around the mean, consistent with this xy point.
ubpdqn suggested several approaches to performing the integration in the square, including:
NIntegrate[PDF[bvn, {x, y}], {x, xl, xu}, {y, yl, yu}]
and:
reg = ImplicitRegion[xl < x < xu && yl < y < yu, {x, y}]
NIntegrate[PDF[bvn, {x, y}], {x, y} \[Element] reg]
I have attempted to modifiy this implicit region such that it becomes an ellipse centered on the mean of the multinormal distribution (-3.89764, 1.29137), and bounded by the xy point like this:
xl = -3.98529;
yl = 1.27792;
xc = -3.89764;
yc = 1.29137;
reg2 = ImplicitRegion[((x - xc)^2/xl^2) + ((y - yc)^2/yl^2) == 1, {x,y}];
NIntegrate[PDF[bvn2, {x, y}], {x, y} \[Element] reg2]
Although this isn't yielding a sensible answer so it is clear to me that I'm doing something wrong. If anyone could suggest a modification to my approach I would be very grateful.