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Re: Unexpected expectation behaviour
*To*: mathgroup at smc.vnet.net
*Subject*: [mg131088] Re: Unexpected expectation behaviour
*From*: Donagh Horgan <donagh.horgan at gmail.com>
*Date*: Tue, 11 Jun 2013 02:28:11 -0400 (EDT)
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On Thursday, June 6, 2013 12:22:51 PM UTC+1, Roland Franzius wrote:
> 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[y-1]((y-1)^3) or Abs[(y-1)^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
Great, thanks. That makes a lot of sense. The numerical method (using NExpectation) appears to be the correct one, so I'll stick with that for now, but it's worth bearing in mind for the future. I tend to think Mathematica is infallible (I know I shouldn't, but it's hard not to), and it's disconcerting to see obvious bugs like this.
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