Template:Gumbel probability paper

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Probability Paper

The form of the Gumbel probability paper is based on a linearization of the [math]\displaystyle{ cdf }[/math] . From Eqn. (UnrGumbel):

[math]\displaystyle{ z=\ln (-\ln (1-F)) }[/math]


using Eqns. (z3):
[math]\displaystyle{ \frac{T-\mu }{\sigma }=\ln (-\ln (1-F)) }[/math]


Then:
[math]\displaystyle{ \ln (-\ln (1-F))=-\frac{\mu }{\sigma }+\frac{1}{\sigma }T }[/math]


Now let:
[math]\displaystyle{ y=\ln (-\ln (1-F)) }[/math]


[math]\displaystyle{ x=T }[/math]


and:
[math]\displaystyle{ \begin{align} & a= & -\frac{\mu }{\sigma } \\ & b= & \frac{1}{\sigma } \end{align} }[/math]


which results in the linear equation of:

[math]\displaystyle{ y=a+bx }[/math]


The Gumbel probability paper resulting from this linearized [math]\displaystyle{ cdf }[/math] function is shown next.

Probabilitygumbel.gif

For [math]\displaystyle{ z=0 }[/math] , [math]\displaystyle{ T=\mu }[/math] and [math]\displaystyle{ R(t)={{e}^{-{{e}^{0}}}}\approx 0.3678 }[/math] (63.21% unreliability). For [math]\displaystyle{ z=1 }[/math] , [math]\displaystyle{ \sigma =T-\mu }[/math] and [math]\displaystyle{ R(t)={{e}^{-{{e}^{1}}}}\approx 0.0659. }[/math] To read [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \mu }[/math] value.