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Re: Bug in multiple integrals with delta function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg90130] Re: Bug in multiple integrals with delta function
  • From: "Dr. Wolfgang Hintze" <weh at snafu.de>
  • Date: Mon, 30 Jun 2008 04:52:33 -0400 (EDT)
  • References: <g3d9me$pse$1@smc.vnet.net>

Hello group,

I'd like to repeat my question. Could someone please comment.
I'm pretty sure that this is a bug, and what's more:
it's fiendish because it starts to appear only from a certain value of 
a "counter" n.

Thanks for your participation.

Regards,
Wolfgang

"Dr. Wolfgang Hintze" <weh at snafu.de> schrieb im Newsbeitrag 
news:g3d9me$pse$1 at smc.vnet.net...
> Consider the following problem: let x1, x2, ..., xn be independent
> random variables uniformly distributed between 0 and 1, and let X =
> Max(x1, ..., xn) and Y = Min(x1,...,xn).
> The problem consists in calculating the distribution function g[n,t] 
> of
> the variable Z = X-Y.
>
> In trying to solve this problem in Mathematica starting with small n
> and guessing the general formula I found for n>=5 some strange and
> buggy behaviour of a multiple integral containing the function
> DiracDelta.
>
> I took the following approach
>
> n=2
> g2 = Integrate[DiracDelta[t - Max[x1, x2] + Min[x1, x2]], {x1, 0, 1},
> {x2, 0, 1}, Assumptions -> {0 < t < 1}]
> Mathematica wouldn't do the integral. Hence I tried to help it by
> splitting the range of integration thus
>
> Integrate[DiracDelta[t - Max[x1, x2] + Min[x1, x2]], {x1, 0, 1}, {x2,
> x1, 1}, Assumptions -> {0 < t < 1}]
> 1-t
>
> and
>
> Integrate[DiracDelta[t - Max[x1, x2] + Min[x1, x2]], {x1, 0, 1}, {x2,
> 0, x1}, Assumptions -> {0 < t < 1}]
> 1 - t
>
> which leads to
>
> g2 = 2(1-t)
>
> n=3
> Splitting the range of integration corresponding to x1<x2<x3 we note
> that this would give us 1/3! of the whole range. Observing 
> furthermore
> that now Max = x3 and Min = x1 we have
>
> g3 = 3!*Integrate[DiracDelta[t - x3 + x1], {x1, 0, 1}, {x2, x1, 1},
> {x3, x2, 1}, Assumptions -> {0 < t < 1}]
> -6*(-1 + t)*t
>
> n=4, similarly
> g4 = 4!*Integrate[DiracDelta[t - x4 + x1], {x1, 0, 1}, {x2, x1, 1},
> {x3, x2, 1}, {x4, x3, 1}, Assumptions -> {0 < t < 1}]
> -12*(-1 + t)*t^2
>
> We would hence guess the general formula to be
>
> (*)    gk = k(k-1) (1-t) t^(k-2)
>
> But let's continue one step further:
>
> n=5, similarly
> g5 = 5!*Integrate[DiracDelta[t - x5 + x1], {x1, 0, 1}, {x2, x1, 1},
> {x3, x2, 1}, {x4, x3, 1}, {x5, x4, 1}, Assumptions -> {0 < t < 1}]
> 10*t^2*(6 - 4*t + t^2)
>
> I don't need to bother you further. Since here is the bug! According 
> to
> (*) the result should be
> g5ok = 20 (1-t) t^3
> The correct result can also be confirmed by a somewhat different 
> method
> in Mathematica, viz.
>
> Integrate[Exp[I*w*(Max[x1, x2, x3, x4, x5] - Min[x1, x2, x3, x4, 
> x5])],
> {x1, 0, 1}, {x2, 0, 1}, {x3, 0, 1}, {x4, 0, 1}, {x5, 0, 1}];
> Simplify[(1/Sqrt[2*Pi])*InverseFourierTransform[%, w, t], 0 < t < 1]
> -20*(-1 + t)*t^3
>
> Hence my question: why does the multiple integral works fine for n=2,
> 3, and 4, but fails for n=5?
>
> Any hint is greatly acknowledged.
>
> Regards,
> Wolfgang
>
> 



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