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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:
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| :* 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.
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| :* 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> .
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| :* As <math>\mu </math> decreases, the <math>pdf</math> is shifted to the left.
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| :* As <math>\mu </math> increases, the <math>pdf</math> is shifted to the right.
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| [[Image:WB.16 gumbel pdf.png|center|400px| ]] | |
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| :* As <math>\sigma </math> increases, the <math>pdf</math> spreads out and becomes shallower.
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| :* As <math>\sigma </math> decreases, the <math>pdf</math> becomes taller and narrower.
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| :* 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>
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| [[Image:WB.16 effect of sigma.png|center|400px| ]]
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| :* 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> .
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| :* 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]]].
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