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Re: Unexpected expectation behaviour
Posted:
Jun 6, 2013 7:22 AM


Am 05.06.2013 09:15, schrieb Donagh Horgan: > Hi all, > > I've been playing around with the following expected value, but I've run into some odd behaviour with Expectation and NExpectation. The following example illustrates the problem: > > Expectation[Abs[y  1]^3, > y \[Distributed] NoncentralChiSquareDistribution[1, s]] > Plot[{%, NExpectation[Abs[y  1]^3, > y \[Distributed] NoncentralChiSquareDistribution[1, s]]}, {s, 0, > 10}] > > All advice greatly appreciated.
Mathematica uses algebraic knowlegde about the parameters in MarcumQ function, the cumulative distribution
CDF[ NoncentralChiSquareDistribution[1, s]]
Take a trace
Trace[Expectation[y  1, y \[Distributed] NoncentralChiSquareDistribution[1, s]]]]
and you will see an evaluation process typical for gaussian integrals.
Unfortunately these procedures do not work for distributional observables like
UnitStep[y1]((y1)^3) or Abs[(y1)^3]
and the like.
In this cases one needs the primitive integrals over the density function explicitely.
So Mathematica gets lost as usual in its unevaluted and never fully understandable evaluation processes of definite integrals.
WRI should change this odd behaviour by just implementing a table lookup eg in Prudnikov/Marichev et al Tables.
The big conceptual error is to let users fill in transformed variables just for fun instead of stating a certain algebraic type of integral and the transformation rules of arguments and parameters applied.
A user friendly definite integrate processor should end the search with a comment and an Abort if the definite integral seems to be unknown.
As a mathematical physicist, I am quite unhappy with WRI's "Integrate policy": Not to cite the sources and hiding algorithmic trivialities clear to the community just for fear of what?
In the case of existing notation differences eg between Abramovitz/Stegun und body of the mathematical literature of function theory  eg in the case of elliptic functions  the Mathematica function body itself needs much more commentaries and hints in which cases to use which functions.

Roland Franzius



