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-Lognormal Reliability Function====
<br>
The reliability for a mission of time  <math>T</math> , starting at age 0, for the Eyring-lognormal model is determined by:
 
<br>
::<math>R(T,V)=\mathop{}_{T}^{\infty }f(t,V)dt</math>
 
<br>
:or:
 
<br>
::<math>R(T,V)=\mathop{}_{{{T}^{^{\prime }}}}^{\infty }\frac{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>
 
<br>
There is no closed form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables. Since the application automatically solves for the reliability we will not discuss manual solution methods.
 
====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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