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| ==Eyring-Lognormal==
| | #REDIRECT [[Eyring_Relationship#Eyring-Lognormal]] |
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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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| <br>
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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.
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| • <math>{{\sigma }_{{{T}'}}}=</math> standard deviation of the natural logarithms of the times-to-failure.
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| <br>
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| The Eyring-lognormal model can be obtained first by setting <math>\breve{T}=L(V)</math>:
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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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| <br>
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| Substituting this into the lognormal <math>pdf</math> yields the Eyring-lognormal model <math>pdf</math>
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| :or:
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| ::<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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| <br>
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| {{eyring-log stat prop sum}}
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| ===Parameter Estimation===
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| {{eyring-log mle}}
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