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Re: Moment's of student's t distribution

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  • Subject: [mg130265] Re: Moment's of student's t distribution
  • From: Roland Franzius <roland.franzius at>
  • Date: Sun, 31 Mar 2013 03:09:50 -0400 (EDT)
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  • References: <kj66eg$ogf$>

Am 30.03.2013 09:06, schrieb paul:
> I'm trying to get a general formula for the moments of a non-standardized t distribution. When I ask for the moments in general --
> Moment[StudentTDistribution[\[Mu], \[Sigma], \[Nu]], k]
> -- Mathematica merely echoes the input. But when I plug in any integer for k, Mathematica gives me a simple answer quickly. How can I get Mathematica to tell me the general formula?

Most "functions" of the statistical apparatus are specialized to be used 
for real valued data by dummies.

Analytically, its always easier to use the standard definition for an 
expectation of function a of a random variable X with cumulative 
distribution F

Mean[a[X]]= Integrate[a[x] D[F[x],x],{x,-inf,inf}]/
Integrate[ D[F[x],x],{x,-inf,inf}]

Analytically Mathematica yields:

Fix real positive parameters:

In[58]:= $Assumptions = \[Mu]/2 > k > 0 &&
    k \[Element] Integers && \[Mu] > 0 && \[Sigma] > 0;

Get the density

  D[CDF[StudentTDistribution[\[Nu], \[Sigma], \[Mu]]][x], x]]

Out[52]= (\[Mu]^(\[Mu]/
   2) \[Sigma]^\[Mu] ((x - \[Nu])^2 + \[Mu] \[Sigma]^2)^(
  1/2 (-1 - \[Mu])))/Beta[\[Mu]/2, 1/2]

Throw away all linear distortion parameters X-> a X + b and 
normalization constants in order to get the pure standard unnormalized 
functional form of the probability density:

In[53]:= f = (#^2 + \[Mu] )^(1/2 (-1 - \[Mu])) &;

Calculate Mean[a[X]]= Int[a dF ]/int[dF]

  Integrate[x^(2 k) f[x], {x, -\[Infinity], \[Infinity]}]/
   Integrate[f[x], {x, -\[Infinity], \[Infinity]}]]

Out[59]= (\[Mu]^k Gamma[1/2 + k] Gamma[-k + \[Mu]/2])/(Sqrt[\[Pi]]


Roland Franzius

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