Re: Mode of InverseChiSquareDistribution

*To*: mathgroup at smc.vnet.net*Subject*: [mg128652] Re: Mode of InverseChiSquareDistribution*From*: Bill Rowe <readnews at sbcglobal.net>*Date*: Wed, 14 Nov 2012 01:28:06 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net

On 11/13/12 at 12:02 AM, paulvonhippel at yahoo.com (paul) wrote: >As a first step toward solving a more complicated problem, I would >like to calculate the mode of the InverseChiSquareDistribution with >D-1 degrees of freedom. The answer is 1/(D+1) but I am having >trouble getting that expression from Mathematica. >First I type >PDF[InverseChiSquareDistribution[D - 1]] >And then I cut and paste the function into ArgMax, imposing >appropriate constraints: >modeInverseChiSquare = ArgMax[{(2^((1 - D)/2) (1/x)^(1 + 1/2 (-1 + >D)) E^(-(1/(2 x))))/ Gamma[1/2 (-1 + D)], x >>0, D > 0, Element[D, Integers]}, x, Reals] >But all ArgMax does is echo the input. If I evaluate the mode at a >particular value of D I get the right answer -- e.g., >modeInverseChiSquare /. D -> 10 returns 1/11. But what I'd like >Mathematica to do is tell me that the answer in general is 1/(D+1). I get the same result with ArgMax. Not sure why. But here is a way to get Mathematica to generate the general result In[1]:= f = Assuming[x > 0, Simplify@PDF[InverseChiSquareDistribution[d - 1], x]]; Quiet@Solve[D[f, x] == 0, x] Out[2]= {{x->1/(d+1)}} Here, I've used Quiet to suppress the warning Solve generates regarding using inverse functions.