Gumbel Distribution

Author

Eric W. Weisstein
March 22, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Statistics/GumbelDistribution.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/GumbelDistribution.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Initialization

If[$VersionNumber<6.0,
GumbelDistribution[0,1]:=ExtremeValueDistribution[0,1]
]

Stats

Stats[GumbelDistribution[0,1],x]//StatsForm

Stats[GumbelDistribution[α,β],x]//StatsForm

PDF

PDF[GumbelDistribution[α,β],x]

Normalization

Integrate[PDF[GumbelDistribution[α,β],x],{x,-∞,∞}]

Integrate[PDF[GumbelDistribution[α,β],x],{x,-∞,∞},Assumptions->(α|β)∈Reals]

Integrate[PDF[GumbelDistribution[α,β],x],{x,-∞,∞},Assumptions->0<α<β]

CDF

CDF[GumbelDistribution[α,β],x]

Integrate[PDF[GumbelDistribution[α,β],x],{x,-∞,xx}]

Integrate[PDF[GumbelDistribution[α,β],x],{x,-∞,xx},Assumptions->0<α<β]

..wrong...

Raw Moments

Integrate[(x^(n ))PDF[GumbelDistribution[α,β],x],{x,-∞,∞},Assumptions->0<α<β]//Timing

Table[Integrate[(x^(n ))PDF[GumbelDistribution[α,β],x],{x,-∞,∞},Assumptions->0<α<β],{n,0,4}]//Expand

Central Moments

Mean[GumbelDistribution[α,β]]

Integrate[(((x-(α-EulerGamma β)))^(n ))PDF[GumbelDistribution[α,β],x],{x,-∞,∞},Assumptions->0<α<β]//Timing

Table[Integrate[(((x-(α-EulerGamma β)))^(n ))PDF[GumbelDistribution[α,β],x],{x,-∞,∞},Assumptions->0<α<β],{n,0,4}]//Expand