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| ==Lognormal Probability Density Function==
| | #REDIRECT [[The Lognormal Distribution]] |
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| The lognormal distribution is a two-parameter distribution with parameters <math>{\mu }'</math> and <math>{{\sigma }_{{{T}'}}}</math> . The <math>pdf</math> for this distribution is given by:
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| ::<math>f({T}')=\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{{T}^{\prime }}-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
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| where, <math>{T}'=\ln (T)</math>. , where the <math>T</math> values are the times-to-failure, and
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| :<math>\mu'=\text{mean of the natural logarithms}</math>
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| :<math>\text{of the times-to-failure,}</math>
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| :<math>\sigma_{T'}=\text{standard deviation of the natural logarithms}</math>
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| :<math>\text{of the times-to-failure}</math>
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| The lognormal <math>pdf</math> can be obtained, realizing that for equal probabilities under the normal and lognormal <math>pdf</math> s, incremental areas should also be equal, or:
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| ::<math>f(T)dT=f({T}')d{T}'</math>
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| Taking the derivative yields:
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| ::<math>d{T}'=\frac{dT}{T}</math>
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| :Substitution yields:
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| ::<math>\begin{align}
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| f(T)= & \frac{f({T}')}{T}, \\
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| f(T)= & \frac{1}{T\cdot {{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(T)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}
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| \end{align}</math>
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| :where:
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| ::<math>f(T)\ge 0,T>0,-\infty <{\mu }'<\infty ,{{\sigma }_{{{T}'}}}>0</math>
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