Template:Gumbel probability paper

Probability Paper
The form of the Gumbel probability paper is based on a linearization of the $$cdf$$. From the unreliabililty equation, we know:


 * $$z=\ln (-\ln (1-F))$$

using the equation for z, we get:


 * $$\frac{t-\mu }{\sigma }=\ln (-\ln (1-F))$$

Then:


 * $$\ln (-\ln (1-F))=-\frac{\mu }{\sigma }+\frac{1}{\sigma }t$$

Now let:


 * $$y=\ln (-\ln (1-F))$$


 * $$x=t$$

and:


 * $$\begin{align}

& a= & -\frac{\mu }{\sigma } \\ & b= & \frac{1}{\sigma } \end{align}$$

which results in the linear equation of:


 * $$y=a+bx$$

The Gumbel probability paper resulting from this linearized $$cdf$$  function is shown next.



For  $$z=0$$,  $$t=\mu $$  and  $$R(t)={{e}^{-{{e}^{0}}}}\approx 0.3678$$  (63.21% unreliability). For  $$z=1$$,  $$\sigma =T-\mu $$  and  $$R(t)={{e}^{-{{e}^{1}}}}\approx 0.0659.$$  To read  $$\mu $$  from the plot, find the time value that corresponds to the intersection of the probability plot with the 63.21% unreliability line. To read $$\sigma $$  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  $$\mu $$  value.