Template:Lognormal distribution probability density function: Difference between revisions

Latest revision as of 04:41, 13 August 2012

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-==Lognormal Probability Density Function==
+#REDIRECT [[The Lognormal Distribution]]
-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:
-::<math>f({T}')=\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{{T}^{\prime }}-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
-where, <math>{T}'=\ln (T)</math>. , where the  <math>T</math>  values are the times-to-failure, and
-:<math>\mu'=\text{mean of the natural logarithms}</math>
-:<math>\text{of the times-to-failure,}</math>
-:<math>\sigma_{T'}=\text{standard deviation of the natural logarithms}</math>
-:<math>\text{of the times-to-failure}</math>
-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:
-::<math>f(T)dT=f({T}')d{T}'</math>
-Taking the derivative yields:
-::<math>d{T}'=\frac{dT}{T}</math>
-:Substitution yields:
-::<math>\begin{align}
-   f(T)= & \frac{f({T}')}{T}, \\
-  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}}}}
-\end{align}</math>
-:where:
-::<math>f(T)\ge 0,T>0,-\infty <{\mu }'<\infty ,{{\sigma }_{{{T}'}}}>0</math>