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 > >