Template:LognormalDistribution: Difference between revisions
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===The Lognormal Distribution=== | === 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. | 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. <br>The <span class="texhtml">''p''''d''''f''</span> of the lognormal distribution is given by: | ||
When the natural logarithms of the times-to-failure are normally distributed, then we say that the data follow the lognormal distribution. | |||
<br> | |||
The < | |||
::<math>\begin{align} | ::<math>\begin{align} | ||
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\end{align} | \end{align} | ||
</math> | </math> | ||
<br> | |||
where, | <br>where, | ||
::<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'}= \text{standard deviation of the natural logarithms of the times to failure} | & {\sigma'}= \text{standard deviation of the natural logarithms of the times to failure} | ||
\end{align}</math> | \end{align}</math> | ||
The lognormal distribution and its characteristics are presented in | The lognormal distribution and its characteristics are presented in detail in the chapter [[The Lognormal Distribution|The Lognormal Distribution]]. | ||
<br> | <br> | ||
Revision as of 20:10, 11 March 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 p'd'f 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{ \begin{align} & {\mu'}= \text{mean of the natural logarithms of the times-to-failure} \\ & {\sigma'}= \text{standard deviation of the natural logarithms of the times to failure} \end{align} }[/math]
The lognormal distribution and its characteristics are presented in detail in the chapter The Lognormal Distribution.