Template:LognormalDistribution

The Lognormal Distribution

The lognormal distribution is commonly used for general reliability analysis, cycles-to-failure in fatigue, material strengths and loading variables in probabilistic design. When the natural logarithms of the times-to-failure are normally distributed, then we say that the data follow the lognormal distribution.
The [math]\displaystyle{ pdf }[/math] of the lognormal distribution is given by:

[math]\displaystyle{ \begin{align} & f(t)=\frac{1}{t{\sigma}_{T'}\sqrt{2\pi}}e^{-\tfrac{1}{2}(\tfrac{T'-{\mu'}}{\sigma_{T'}})^2}\\ & f(t)\ge 0,t\gt 0,{{\sigma }_{T'}}\gt 0 \\ & {T'}= \ln (t) \end{align} }[/math]

Where,

[math]\displaystyle{ \begin{align} & {\mu'}= \text{mean of the natural logarithms of the times-to-failure} \\ & {\sigma_{T'}}= \text{standard deviation of the natural logarithms of the times to failure} \end{align} }[/math]

The lognormal distribution and its characteristics are presented in more detail in Chapter 10.