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 <span class="texhtml">''pdf''</span> of the lognormal distribution is given by:  
The <math>pdf</math> of the lognormal distribution is given by:  


::<math>\begin{align}
::<math>\begin{align}
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</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]].

Revision as of 03:01, 22 August 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 [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.