Template:Erying-log

Eyring-Lognormal
The $$pdf$$  of the lognormal distribution is given by:


 * $$f(T)=\frac{1}{T\text{ }{{\sigma }_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline}{{{\sigma }_}} \right)}^{2}}}}$$

where:


 * $${T}'=\ln (T)$$


 * $$T=\text{times-to-failure}$$

and •	 $$\overline=$$ mean of the natural logarithms of the times-to-failure.

•	 $${{\sigma }_}=$$ standard deviation of the natural logarithms of the times-to-failure.

The Eyring-lognormal model can be obtained first by setting $$\breve{T}=L(V)$$:


 * $$\breve{T}=L(V)=\frac{1}{V}{{e}^{-(A-\tfrac{B}{V})}}$$


 * or:


 * $${{e}^}=\frac{1}{V}{{e}^{-(A-\tfrac{B}{V})}}$$


 * Thus:


 * $${{\overline{T}}^{\prime }}=-\ln (V)-A+\frac{B}{V}$$

Substituting this into the lognormal $$pdf$$ yields the Eyring-lognormal model $$pdf$$:


 * $$f(T,V)=\frac{1}{T\text{ }{{\sigma }_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_}} \right)}^{2}}}}$$