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MathGroup Archive 1993

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Failure to recognize "1"

  • To: mathgroup at yoda.physics.unc.edu
  • Subject: Failure to recognize "1"
  • From: mek at guinan.psu.edu (Mark E. Kotanchek)
  • Date: Wed, 8 Sep 93 21:57:54 -0400

Hi,

I'm currently working with the average of the autocovariance of a Gaussian  
random variable. Hence, I defined the function

fChi2[R2_,sigma_,N_] := PDF[ChiSquareDistribution[N],R2/(sigma^2)]/sigma^2

which is the density function for a chi-squared distribution in the  
non-normalized case. (The Mma function presumes a standard deviation of 1.)  
Anyhow, given my hard won suspicion of Mma integrals, I checked the expected  
value and standard deviation. Running Mma 2.1 on my NeXTstation I got,

In[39]:=
   Declare[sigma, NonNegative]
   Declare[sigma, Real]
   Declare[M, Positive]
   expectedAvg = Integrate[R2 fChi2[R2,sigma,M],{R2,0,Infinity}]
   secondMoment = Integrate[R2^2 fChi2[R2,sigma,M],{R2,0,Infinity}]
   stdDeviation = Sqrt[secondMoment - expectedAvg^2]
   

Out[37]=
           2
    M sigma
   -----------   <------ should be "M sigma"
    1 M/2  M/2
   (-)    2
    2
   

Out[38]=
                  4
   M (2 + M) sigma
   ----------------
      1 M/2  M/2
     (-)    2
      2
   

Out[39]=
           2      4                   4
          M  sigma     M (2 + M) sigma
   Sqrt[-(---------) + ----------------] <-- Sqrt[2 M] sigma^2
            1 M  M        1 M/2  M/2
           (-)  2        (-)    2
            2             2

My question is why Mma doesn't recognize that

      1 M/2  M/2
     (-)    2      = "1"?
      2
      

As a result, the above equation doesn't simplify upon application of simplify  
or expand. Is there some subtlety I'm missing or is this a bug in Mma?

Thanks,

Mark.
---
Mark Kotanchek
Guidance & Control Dept - 363 ASB
Applied Research Lab/Penn State
P.O. Box 30
State College, PA 16804

e-mail:	mek at guinan.psu.edu (NeXTmail)
TEL:	(814)863-0682
FAX:	(814)863-7843





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