Bug in multiple integrals with delta function

*To*: mathgroup at smc.vnet.net*Subject*: [mg89729] Bug in multiple integrals with delta function*From*: "Dr. Wolfgang Hintze" <weh at snafu.de>*Date*: Thu, 19 Jun 2008 05:42:50 -0400 (EDT)

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