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: | 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> | <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: | where: | ||
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<math>{T}'=\ln (T)</math> | <math>{T}'=\ln (T)</math> | ||
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<math>T=\text{times-to-failure}</math> | <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>\overline{{{T}'}}=</math> mean 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: | 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> | <math>\breve{T}=L(V)=\frac{1}{V}{{e}^{-(A-\tfrac{B}{V})}}</math> | ||
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or: | or: | ||
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<math>{{\overline{T}}^{\prime }}=-\ln (V)-A+\frac{B}{V}</math> | <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> | Substituting Eqn. (eyr-logn-mean) into Eqn. (Eyr-logn-pdf) yields the Eyring-lognormal model <math>pdf</math> | ||
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Revision as of 21:51, 10 February 2012
Standard Folio Data Eyring-Lognormal |
ALTA |
The [math]\displaystyle{ pdf }[/math] of the lognormal distribution is given by:
• [math]\displaystyle{ {{\sigma }_{{{T}'}}}= }[/math] standard deviation of the natural logarithms of the times-to-failure.
[math]\displaystyle{ 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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