Template:LognormalDistribution: Difference between revisions
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::<math>\begin{align} | ::<math>\begin{align} | ||
& f(t)=\frac{1}{t{\sigma}_{ | & 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 }_{ | & f(t)\ge 0,t>0,{{\sigma }_{t'}}>0 \\ | ||
& { | & {t'}= \ln (t) | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
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::<math>\begin{align} | ::<math>\begin{align} | ||
& {\mu'}= \text{mean of the natural logarithms of the times-to-failure} \\ | & {\mu'}= \text{mean of the natural logarithms of the times-to-failure} \\ | ||
& {\sigma_{ | & {\sigma_{t'}}= \text{standard deviation of the natural logarithms of the times to failure} | ||
\end{align}</math> | \end{align}</math> | ||
Revision as of 21:20, 3 February 2012
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.