NExpectation not up to expectations with Boole or Conditioned

Question

Context

I am interested in computing numerically the number of extrema at a given threshold for random fields. These numbers are expectations of MultinormalDistributions.

Problem

This integral should give the differential number of maxima of a scale invariant Gaussian random field ($\gamma=0$ i.e. white noise) at threshold zero.

NExpectation[(-(x*y) + z^2)*Boole[x*y - z^2 <= 0], 
{x, z, y} \[Distributed] 
MultinormalDistribution[{0, 0, 0}, {{3/8, 0, 1/8}, {0, 1/8, 0}, {1/8, 0, 3/8}}]]

Its value should be 1/4/Sqrt[3]= 0.144338 whereas Mathematica returns 0.138245 which is wrong by a factor of 5 %.

Strangely enough if I split the integral in two as

NExpectation[(-(x*y) + z^2)*Boole[x + y >= 0 && x*y - z^2 <= 0], 
 {x, z, y} \[Distributed] 
 MultinormalDistribution[{0, 0, 0},{{3/8, 0, 1/8}, {0, 1/8, 0}, {1/8, 0, 3/8}}]] +
NExpectation[(-(x*y) + z^2)*Boole[x + y <= 0 && x*y - z^2 <= 0], 
  {x, z, y} \[Distributed] 
MultinormalDistribution[{0, 0, 0}, {{3/8, 0, 1/8}, {0, 1/8, 0}, {1/8, 0, 3/8}}]]

Mathematica returns the correct answer to 5 digits.

I understand integrating numerically Boole is likely to cause problems but I am nonetheless surprised by the poor result.

On a related topic

 NExpectation[-(x*y) + z^2 \[Conditioned] (x*y - z^2 < 0), 
 {x, z, y} \[Distributed] 
 MultinormalDistribution[{0, 0, 0}, {{3/8, 0, 1/8}, {0, 1/8, 0}, {1/8, 0, 3/8}}]]

seems to take a while before returning unevaluated (even though it should probably be the correct way to proceed?).

Questions

1. Why is the first method doing so poorly?
2. Why is the second method failing altogether?
3. Any guidance on how to tell NExpectation to be "smart" about the Boole condition?

Thanks

Regards

Answer 1

Numerical integration in Mma is a big topic. I suggest reading at least this.

The following gives the correct answer:

NExpectation[(-(x*y) + z^2) Boole[x*y - z^2 <= 0], {x, z, y} \[Distributed] 
  MultinormalDistribution[{0, 0, 0}, {{3/8, 0, 1/8}, {0, 1/8, 0}, {1/8, 0, 3/8}}], 
     Method -> {"NIntegrate", {Exclusions -> True}}]