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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<br>
<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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<br>
where:  
where:  
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<br>
<math>{T}'=\ln (T)</math>
<math>{T}'=\ln (T)</math>
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<br>
<math>T=\text{times-to-failure}</math>
<math>T=\text{times-to-failure}</math>
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:and:
and:
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<br>
• <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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<br>
<br>
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:
<br>
<br>
<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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<br>
or:
or:
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<br>
<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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<br>
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

Webnotes-alta.png
Standard Folio Data Eyring-Lognormal
ALTA

The [math]\displaystyle{ pdf }[/math] of the lognormal distribution is given by:
[math]\displaystyle{ 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]
where:
[math]\displaystyle{ {T}'=\ln (T) }[/math]
[math]\displaystyle{ T=\text{times-to-failure} }[/math]
and:
[math]\displaystyle{ \overline{{{T}'}}= }[/math] mean of the natural logarithms of the times-to-failure.

[math]\displaystyle{ {{\sigma }_{{{T}'}}}= }[/math] standard deviation of the natural logarithms of the times-to-failure.


The Eyring-lognormal model can be obtained first by setting [math]\displaystyle{ \breve{T}=L(V) }[/math] in Eqn. (eyring). Therefore:
[math]\displaystyle{ \breve{T}=L(V)=\frac{1}{V}{{e}^{-(A-\tfrac{B}{V})}} }[/math]
or:

[math]\displaystyle{ {{e}^{{{\overline{T}}^{\prime }}}}=\frac{1}{V}{{e}^{-(A-\tfrac{B}{V})}} }[/math]


Thus:

[math]\displaystyle{ {{\overline{T}}^{\prime }}=-\ln (V)-A+\frac{B}{V} }[/math]
Substituting Eqn. (eyr-logn-mean) into Eqn. (Eyr-logn-pdf) yields the Eyring-lognormal model [math]\displaystyle{ pdf }[/math] or:

[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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