Template:Lognormal Distribution Definition: Difference between revisions
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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 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 | The ''pdf'' of the lognormal distribution is given by: | ||
::<math>\begin{align} | ::<math>\begin{align} | ||
| Line 7: | Line 7: | ||
& f(t)\ge 0,t>0,{\sigma'}>0 \\ | & f(t)\ge 0,t>0,{\sigma'}>0 \\ | ||
& {t'}= \ln (t) | & {t'}= \ln (t) | ||
\end{align} | \end{align}\,\! | ||
</math> | </math> | ||
where <math>{\mu'}</math> is the mean of the natural logarithms of the times-to-failure and <math>{\sigma'}</math> is the standard deviation of the natural logarithms of the times to failure. | where <math>{\mu'}\,\!</math> is the mean of the natural logarithms of the times-to-failure and <math>{\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]]. | For a detailed discussion of this distribution, see [[The Lognormal Distribution]]. | ||
Latest revision as of 21:45, 21 September 2012
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:
- [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.