How is P(Y=0) *<=* P((Y-EY)^2=(EY)^2), would *=* be more precise ?

]]>It doesn’t give an answer to “why”, but at least gives a non-mysterious “how”. ]]>

I think you mean things like Gaussian quadrature rules? As far as I know, the issue with them is that (1) they work best if the function being integrated has a lot of structure, for example is close to a polynomial; (2) there are no sufficiently accurate variants for higher dimensions. However, if you have a one dimensional integral and you think your function is close to a polynomial, a Gaussian quadrature rule might be the right thing to use.

Sasho

]]>