       Re: power of logistic distribution

• To: mathgroup at smc.vnet.net
• Subject: [mg126918] Re: power of logistic distribution
• From: Bob Hanlon <hanlonr357 at gmail.com>
• Date: Mon, 18 Jun 2012 05:41:00 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201206170758.DAA08867@smc.vnet.net>

```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 <paulvonhippel at yahoo.com> 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:= Mean[TransformedDistribution[Z^2,
>   Z \[Distributed] LogisticDistribution[\[Mu], \[Sigma]]]]
> Out= 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:= Assuming[p \[Element] Integers,
>  Mean[TransformedDistribution[Z^p,
>    Z \[Distributed] LogisticDistribution[\[Mu], \[Sigma]]]]]
> Out= Mean[
> TransformedDistribution[\[FormalX]^
>  p, \[FormalX] \[Distributed] LogisticDistribution[\[Mu], \[Sigma]]]]
>
> Why isn't it giving me something more digested?
>

```

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