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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'</math> . The <math>pdf</math> for this distribution is given by:
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| ::<math>f({t}')=\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{{t}^{\prime }}-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}</math>
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| where,
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| :<math>{t}'=\ln (t)</math>. <math>t</math> values are the times-to-failure, and
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| :<math>\mu'=\text{mean of the natural logarithms of the times-to-failure,}</math>
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| :<math>\sigma'=\text{standard deviation of the natural logarithms 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' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(t)-{\mu }'}{{{\sigma' }}} \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' }}>0</math>
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