|
|
| (3 intermediate revisions by 2 users not shown) |
| Line 1: |
Line 1: |
| ==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 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: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.
| |