Template:Lognormal Distribution Definition

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

where [math]\displaystyle{ {\mu'}\,\! }[/math] is the mean of the natural logarithms of the times-to-failure and [math]\displaystyle{ {\sigma'}\,\! }[/math] is the standard deviation of the natural logarithms of the times to failure.

For a detailed discussion of this distribution, see The Lognormal Distribution.