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Re: Expected value of the Geometric distribution

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  Re: Expected value of the Geometric distribution Andy Ross <andyr@wolfram.com> - 2011-05-04 10:35 +0000

#2093 — Re: Expected value of the Geometric distribution

On 5/3/2011 7:22 AM, 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
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If you Integrate with the assumptions of the distribution you should get 
what you are expecting...

Integrate[(E^(-E^((x - \[Alpha])/\[Beta]) + (x - 
\[Alpha])/\[Beta])*x)/\[Beta],{x,-\[Infinity],\[Infinity]},Assumptions->DistributionParameterAssumptions[GumbelDistribution[\[Alpha],\[Beta]]]]

-Andy

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