ALTA ALTA Standard Folio Data Eyring-Lognormal: Difference between revisions

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The  <math>pdf</math>  of the lognormal distribution is given by:
 
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<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
 
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where:
 
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<math>{T}'=\ln (T)</math>
 
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<math>T=\text{times-to-failure}</math>
 
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:and:
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• <math>\overline{{{T}'}}=</math> mean of the natural logarithms of the times-to-failure.
 
• <math>{{\sigma }_{{{T}'}}}=</math> standard deviation of the natural logarithms of the times-to-failure.
 
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The Eyring-lognormal model can be obtained first by setting  <math>\breve{T}=L(V)</math> in Eqn. (eyring). Therefore:
 
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<math>\breve{T}=L(V)=\frac{1}{V}{{e}^{-(A-\tfrac{B}{V})}}</math>
 
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or:
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<math>{{e}^{{{\overline{T}}^{\prime }}}}=\frac{1}{V}{{e}^{-(A-\tfrac{B}{V})}}</math>
 
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Thus:
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<math>{{\overline{T}}^{\prime }}=-\ln (V)-A+\frac{B}{V}</math>
 
 
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Substituting Eqn. (eyr-logn-mean) into Eqn. (Eyr-logn-pdf) yields the Eyring-lognormal model  <math>pdf</math> 
or:
 
<math>f(T,V)=\frac{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}}}}</math>
 
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Latest revision as of 18:56, 8 July 2015

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