Re: Moment's of student's t distribution

• To: mathgroup at smc.vnet.net
• Subject: [mg130265] Re: Moment's of student's t distribution
• From: Roland Franzius <roland.franzius at uos.de>
• Date: Sun, 31 Mar 2013 03:09:50 -0400 (EDT)
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• References: <kj66eg\$ogf\$1@smc.vnet.net>

```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

In[52]:=
FullSimplify[
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]

In[59]:=
FullSimplify[
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]]
Gamma[\[Mu]/2])

--

Roland Franzius

```

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