Template:Gumbel probability paper: Difference between revisions

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==Probability Paper==
#REDIRECT [[The_Gumbel/SEV_Distribution#Probability_Paper]]
The form of the Gumbel probability paper is based on a linearization of the  <math>cdf</math> . From the unreliabililty equation, we know:
 
::<math>z=\ln (-\ln (1-F))</math>
 
using the equation for ''z'', we get:
 
::<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:WB.16 probability gumbel.png|center|250px| ]]  
 
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.

Revision as of 03:38, 15 August 2012

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