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 $$pdf$$ of the lognormal distribution is given by:


 * $$\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>0,{{\sigma }_{t'}}>0 \\ & {t'}= \ln (t) \end{align} $$ where,
 * $$\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}$$

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