Template:Eyring-log stat prop sum: Difference between revisions

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===Eyring-Lognormal Statistical Properties Summary===
#REDIRECT [[Eyring_Relationship#Eyring-Lognormal]]
 
{{eyring-log mean}}
 
{{eyring-log median}}
 
{{eyring-log sd}}
 
{{eyring-log mode}}
 
{{eyring-log rf}}
 
====Reliable Life====
<br>
For the Eyring-lognormal model, the reliable life, or the mission duration for a desired reliability goal,  <math>{{t}_{R}},</math>  is estimated by first solving the reliability equation with respect to time, as follows:
 
<br>
::<math>T_{R}^{\prime }=-\ln (V)-A+\frac{B}{V}+z\cdot {{\sigma }_{{{T}'}}}</math>
 
<br>
:where:
 
<br>
::<math>z={{\Phi }^{-1}}\left[ F\left( T_{R}^{\prime },V \right) \right]</math>
 
<br>
:and:
 
<br>
::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}',V)}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>
<br>
<br>
Since  <math>{T}'=\ln (T)</math>  the reliable life,  <math>{{t}_{R,}}</math>  is given by:
 
<br>
::<math>{{t}_{R}}={{e}^{T_{R}^{\prime }}}</math>
 
====Eyring-Lognormal Failure Rate====
 
<br>
The Eyring-lognormal failure rate is given by:
 
::<math>\lambda (T,V)=\frac{f(T,V)}{R(T,V)}=\frac{\tfrac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}}{\mathop{}_{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt}</math>

Latest revision as of 23:32, 16 August 2012

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