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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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