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===Arrhenius-Lognormal Statistical Properties Summary===
#REDIRECT [[Arrhenius_Relationship#Arrhenius-Lognormal_Statistical_Properties_Summary]]
====The Mean====
<br>
• The mean life of the Arrhenius-lognormal model (mean of the times-to-failure),  <math>\bar{T}</math> , is given by:
 
<br>
::<math>\begin{align}
  & \bar{T}= & {{e}^{\bar{{T}'}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}} \\
& = & {{e}^{\ln (C)+\tfrac{B}{V}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}} 
\end{align}</math>
 
<br>
• The mean of the natural logarithms of the times-to-failure,  <math>{{\bar{T}}^{^{\prime }}}</math> , in terms of  <math>\bar{T}</math>  and  <math>{{\sigma }_{T}}</math>  is given by:
 
<br>
::<math>{{\bar{T}}^{\prime }}=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)</math>
 
====The Standard Deviation====
<br>
• The standard deviation of the Arrhenius-lognormal model (standard deviation of the times-to-failure),  <math>{{\sigma }_{T}}</math> , is given by:
 
<br>
::<math>\begin{align}
  & {{\sigma }_{T}}= & \sqrt{\left( {{e}^{2\bar{{T}'}+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} \\
& = & \sqrt{\left( {{e}^{2\left( \ln (C)+\tfrac{B}{V} \right)+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} 
\end{align}</math>
 
<br>
• The standard deviation of the natural logarithms of the times-to-failure,  <math>{{\sigma }_{{{T}'}}}</math> , in terms of  <math>\bar{T}</math>  and  <math>{{\sigma }_{T}}</math>  is given by:
<br>
::<math>{{\sigma }_{{{T}'}}}=\sqrt{\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)}</math>
<br>
====The Mode====
<br>
• The mode of the Arrhenius-lognormal model is given by:
::<math>\begin{align}
  & \tilde{T}= & {{e}^{{{\overline{T}}^{\prime }}-\sigma _{{{T}'}}^{2}}} \\
& = & {{e}^{\ln (C)+\tfrac{B}{V}-\sigma _{{{T}'}}^{2}}} 
\end{align}</math>
====Arrhenius-Lognormal Reliability Function====
<br>
The reliability for a mission of time  <math>T</math> , starting at age 0, for the Arrhenius-lognormal model is determined by:
<br>
::<math>R(T,V)=\mathop{}_{T}^{\infty }f(t,V)dt</math>
 
vor:
<br>
::<math>R(T,V)=\mathop{}_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (C)-\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.
<br>
====Reliable Life====
<br>
For the Arrhenius-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>
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>
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>
 
====Arrhenius-Lognormal Failure Rate====
<br>
The Arrhenius-lognormal failure rate is given by:
<br>
::<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 (C)-\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 (C)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt}</math>
 
====When Using the Lognormal Distribution in ALTA====
<br>
The parameters returned for the Arrhenius-lognormal distribution are always  <math>{{\sigma }_{{{T}'}}},</math>  <math>C,</math>  and  <math>B.</math>  The returned  <math>{{\sigma }_{{{T}'}}}</math>  is always the square root of the variance of the natural logarithms to failure. Also, if the "Show Scale Parameter" option is checked (on the Data Sheet tab in the User Setup), the returned mean value is always the mean of the natural logarithms of the times-to-failure, given by Eqn. (arrh-logn-mean). Even though the application denotes these values as mean and standard deviation, the user is reminded that these are given as parameters of the distribution, and are thus the mean (a function of stress as it can be seen in Eqn. (arrh-logn-mean)) and standard deviation of the natural logarithms of the data. The mean life value of the times-to-failure, as well as the standard deviation of times-to-failure (not the parameter) can be obtained through the Function Wizard in ALTA.
<br>
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Latest revision as of 05:30, 16 August 2012

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