Re: power of logistic distribution

• To: mathgroup at smc.vnet.net
• Subject: [mg126948] Re: power of logistic distribution
• From: paul <paulvonhippel at yahoo.com>
• Date: Tue, 19 Jun 2012 03:16:37 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201206170758.DAA08867@smc.vnet.net> <jrmt45\$h0d\$1@smc.vnet.net>

On Monday, June 18, 2012 4:41:25 AM UTC-5, Bob Hanlon wrote:
> dist = LogisticDistribution[m, s];
>
> Simplify[Moment[dist, n], n >= 0]
>
> (2*Pi)^n*((-I)*s)^n*BernoulliB[n, 1/2 + (I*m)/(2*Pi*s)]
>
> And@@Table[Moment[dist, n] == Mean[TransformedDistribution[z^n, z\[Distributed]dist]], {n, 0, 40}]//Simplify
>
> True
>
>
> Bob Hanlon
>
>
> On Jun 17, 2012, at 3:58 AM, paul
>  wrote:
>
> > I would like an expression for the mean of a variable that is some integer power of a logistic variable. I have tried the following approach, which did not work. Many thanks for any suggestions.
> >
> > Here is what I've done so far. If I specify the power (e.g., power=2), Mathematica returns an answer rather quickly -- e.g.,
> >
> > In[66]:= Mean[TransformedDistribution[Z^2,
> >   Z \[Distributed] LogisticDistribution[\[Mu], \[Sigma]]]]
> > Out[66]= 1/3 (3 \[Mu]^2 + \[Pi]^2 \[Sigma]^2)
> >
> > I get quick results if the power is 2, 3, 4, ..., 100. So it seems to me there must be some general solution for integer powers. But when I ask for that general solution, Mathematica simply echoes the input:
> >
> > In[64]:= Assuming[p \[Element] Integers,
> >  Mean[TransformedDistribution[Z^p,
> >    Z \[Distributed] LogisticDistribution[\[Mu], \[Sigma]]]]]
> > Out[64]= Mean[
> > TransformedDistribution[\[FormalX]^
> >  p, \[FormalX] \[Distributed] LogisticDistribution[\[Mu], \[Sigma]]]]
> >
> > Why isn't it giving me something more digested?
> >

Excellent -- thank you! Two further questions, if I may:
1. Why did the Moment function work where the Mean function did not?
2. As the moment is real, I wonder if there is an expression for it that does not involve complex numbers.

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