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| ===The Lognormal Distribution===
| | #REDIRECT [[The_Lognormal_Distribution]] |
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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.
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| When the natural logarithms of the times-to-failure are normally distributed, then we say that the data follow the lognormal distribution.
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| <br>
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| The <math>pdf</math> of the lognormal distribution is given by:
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| ::<math>\begin{align}
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| & f(t)=\frac{1}{t{\sigma}_{t'}\sqrt{2\pi}}e^{-\tfrac{1}{2}(\tfrac{t'-{\mu'}}{\sigma_{t'}})^2}\\
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| & f(t)\ge 0,t>0,{{\sigma }_{t'}}>0 \\
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| & {t'}= \ln (t)
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| \end{align}
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| </math>
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| <br>
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| where,
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| ::<math>\begin{align}
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| & {\mu'}= \text{mean of the natural logarithms of the times-to-failure} \\
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| & {\sigma_{t'}}= \text{standard deviation of the natural logarithms of the times to failure}
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| \end{align}</math>
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| The lognormal distribution and its characteristics are presented in more detail in [[The Lognormal Distribution |Chapter 10]].
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| <br>
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