Re: Expected value of the Geometric distribution
- To: mathgroup at smc.vnet.net
- Subject: [mg118627] Re: Expected value of the Geometric distribution
- From: Peter Breitfeld <phbrf at t-online.de>
- Date: Wed, 4 May 2011 19:48:30 -0400 (EDT)
- References: <ipos29$i2t$1@smc.vnet.net>
"Tonja Krueger" wrote:
> Dear everybody,
> Thank you all for your kind help. But I'm still stuck trying to find
> the expected value for a continuous distribution like the Gumbel
> distribution or GEV, Weibull.
> Moment[GumbelDistribution[\[Alpha], \[Beta]], 1]
> gives this as result:
> \[Alpha] - EulerGamma \[Beta]
> But when I try using
> Integrate[ E^(-E^(-((x - \[Mu])/\[Beta])) - (x -
> \[Mu])/\[Beta])/\[Beta]* x, {x, -\[Infinity], \[Infinity]}]
> This is what I get:
> ConditionalExpression[\[Beta] (EulerGamma + Log[E^(\[Mu]/\[Beta])] -
> E^-E^((\[Mu]/\[Beta])) Log[E^(-(\[Mu]/\[Beta]))] +
> Log[E^(\[Mu]/\[Beta])])), Re[\[Beta]] > 0]
> I am stumped.
> Tonja
Because GumpelDistribution ist defined for real mu and positive beta you
should take these Assumptions in Integrate:
Integrate[
E^(-E^(-((x - mu)/beta)) - (x - mu)/beta)/beta* x,
{x, -Infinity, Infinity},
Assumptions -> {mu] \[Element] Reals, beta > 0}]
Out= EulerGamma beta + mu
Alternatively, you can Simplify your ConditionalExpression (I'll name it
int)
simpbeta=Simplify[int,beta>0]
Out
beta (EulerGamma + Log[E^(mu/beta)] -
E^-E^((mu/beta)) (Log[E^(-(mu/beta))] + Log[E^(mu/beta)]))
Mathematica assumes all variables to be potentially complex, so Log[E^x]
is not automatically simplified to x because Log is a multivalued
function. You can in this case either use
PowerExpand[simpbeta]
or declare mu to be real
Simplify[int,beta>0 && Element[mu,Reals]]
This will give you the desired result.
--
_________________________________________________________________
Peter Breitfeld, Bad Saulgau, Germany -- http://www.pBreitfeld.de