Template:Gumbel probability paper: Difference between revisions

Latest revision as of 03:38, 15 August 2012

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-==Probability Paper==
+#REDIRECT [[The_Gumbel/SEV_Distribution]]
-The form of the Gumbel probability paper is based on a linearization of the  <math>cdf</math> . From Eqn. (UnrGumbel):
-::<math>z=\ln (-\ln (1-F))</math>
-:using Eqns. (z3):
-::<math>\frac{T-\mu }{\sigma }=\ln (-\ln (1-F))</math>
-:Then:
-::<math>\ln (-\ln (1-F))=-\frac{\mu }{\sigma }+\frac{1}{\sigma }T</math>
-:Now let:
-::<math>y=\ln (-\ln (1-F))</math>
-::<math>x=T</math>
-:and:
-::<math>\begin{align}
-  & a= & -\frac{\mu }{\sigma } \\
- & b= & \frac{1}{\sigma }
-\end{align}</math>
-which results in the linear equation of:
-::<math>y=a+bx</math>
-The Gumbel probability paper resulting from this linearized  <math>cdf</math>  function is shown next.
-[[Image:probabilitygumbel.gif|thumb|center|400px| ]]
-For   <math>z=0</math> ,  <math>T=\mu </math>  and  <math>R(t)={{e}^{-{{e}^{0}}}}\approx 0.3678</math>  (63.21% unreliability). For   <math>z=1</math> ,  <math>\sigma =T-\mu </math>  and  <math>R(t)={{e}^{-{{e}^{1}}}}\approx 0.0659.</math>  To read  <math>\mu </math>  from the plot, find the time value that corresponds to the intersection of the probability plot with the 63.21% unreliability line. To read  <math>\sigma </math>  from the plot, find the time value that corresponds to the intersection of the probability plot with the 93.40% unreliability line, then take the difference between this time value and the  <math>\mu </math>  value.