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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.
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