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==Characteristics of the Gumbel Distribution==
#REDIRECT [[The_Gumbel/SEV_Distribution]]
Some of the specific characteristics of the Gumbel distribution are the following:
 
:* The shape of the Gumbel distribution is skewed to the left. The Gumbel  <math>pdf</math>  has no shape parameter. This means that the Gumbel  <math>pdf</math>  has only one shape, which does not change.
:* The Gumbel  <math>pdf</math>  has location parameter  <math>\mu ,</math>  which is equal to the mode  <math>\tilde{T},</math>  but it differs from median and mean. This is because the Gumbel distribution is not symmetrical about its  <math>\mu </math> .
:* As  <math>\mu </math>  decreases, the  <math>pdf</math>  is shifted to the left.
:* As  <math>\mu </math>  increases, the  <math>pdf</math>  is shifted to the right.
 
[[Image:effectofmuongumbel.gif|thumb|center|400px| ]]
 
:* As  <math>\sigma </math>  increases, the  <math>pdf</math>  spreads out and becomes shallower.
:* As  <math>\sigma </math>  decreases, the  <math>pdf</math>  becomes taller and narrower.
:* For  <math>T=\pm \infty ,</math>  <math>pdf=0.</math>  For  <math>T=\mu </math> , the  <math>pdf</math>  reaches its maximum point <math>\frac{1}{\sigma e}</math>
 
[[Image:effectofsigmaongumbel.gif|thumb|center|400px| ]]
 
:* The points of inflection of the  <math>pdf</math>  graph are  <math>T=\mu \pm \sigma \ln (\tfrac{3\pm \sqrt{5}}{2})</math>  or  <math>T\approx \mu \pm \sigma 0.96242</math> .
:* If times follow the Weibull distribution, then the logarithm of times follow a Gumbel distribution. If  <math>{{t}_{i}}</math>  follows a Weibull distribution with  <math>\beta </math>  and  <math>\eta </math>  , then the  <math>Ln({{t}_{i}})</math>  follows a Gumbel distribution with  <math>\mu =\ln (\eta )</math>  and  <math>\sigma =\tfrac{1}{\beta }</math>  [[Appendix: Weibull References|[32]]] <math>.</math>

Latest revision as of 07:43, 8 August 2012

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