Template:Lognormal distribution probability density function: Difference between revisions

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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:  
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:  


::<math>f({t}')=\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{{T}^{\prime }}-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}</math>
::<math>f({t}')=\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{{t}^{\prime }}-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}</math>


where, <math>{T}'=\ln (T)</math>. , where the  <math>T</math>  values are the times-to-failure, and  
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>\mu'=\text{mean of the natural logarithms}</math>
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:<math>\text{of the times-to-failure,}</math>
:<math>\text{of the times-to-failure,}</math>


:<math>\sigma_{T'}=\text{standard deviation of the natural logarithms}</math>  
:<math>\sigma'=\text{standard deviation of the natural logarithms}</math>  


:<math>\text{of the times-to-failure}</math>
:<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:  
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>
::<math>f(t)dt=f({t}')d{t}'</math>


Taking the derivative yields:  
Taking the derivative yields:  


::<math>d{T}'=\frac{dT}{T}</math>
::<math>d{t}'=\frac{dt}{t}</math>


:Substitution yields:  
:Substitution yields:  


::<math>\begin{align}
::<math>\begin{align}
   f(T)= & \frac{f({T}')}{T}, \\  
   f(t)= & \frac{f({t}')}{t}, \\  
   f(T)= & \frac{1}{T\cdot {{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(T)-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}   
   f(t)= & \frac{1}{t\cdot {{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(t)-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}   
\end{align}</math>
\end{align}</math>


:where:  
:where:  


::<math>f(T)\ge 0,T>0,-\infty <{\mu }'<\infty ,{{\sigma' }}>0</math>
::<math>f(t)\ge 0,T>0,-\infty <{\mu }'<\infty ,{{\sigma' }}>0</math>

Revision as of 16:11, 13 February 2012

Lognormal Probability Density Function

The lognormal distribution is a two-parameter distribution with parameters [math]\displaystyle{ {\mu }' }[/math] and [math]\displaystyle{ \sigma' }[/math] . The [math]\displaystyle{ pdf }[/math] for this distribution is given by:

[math]\displaystyle{ f({t}')=\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{{t}^{\prime }}-{\mu }'}{{{\sigma' }}} \right)}^{2}}}} }[/math]

where, [math]\displaystyle{ {t}'=\ln (t) }[/math]. , where the [math]\displaystyle{ t }[/math] values are the times-to-failure, and

[math]\displaystyle{ \mu'=\text{mean of the natural logarithms} }[/math]
[math]\displaystyle{ \text{of the times-to-failure,} }[/math]
[math]\displaystyle{ \sigma'=\text{standard deviation of the natural logarithms} }[/math]
[math]\displaystyle{ \text{of the times-to-failure} }[/math]

The lognormal [math]\displaystyle{ pdf }[/math] can be obtained, realizing that for equal probabilities under the normal and lognormal [math]\displaystyle{ pdf }[/math] s, incremental areas should also be equal, or:

[math]\displaystyle{ f(t)dt=f({t}')d{t}' }[/math]

Taking the derivative yields:

[math]\displaystyle{ d{t}'=\frac{dt}{t} }[/math]
Substitution yields:
[math]\displaystyle{ \begin{align} f(t)= & \frac{f({t}')}{t}, \\ f(t)= & \frac{1}{t\cdot {{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(t)-{\mu }'}{{{\sigma' }}} \right)}^{2}}}} \end{align} }[/math]
where:
[math]\displaystyle{ f(t)\ge 0,T\gt 0,-\infty \lt {\mu }'\lt \infty ,{{\sigma' }}\gt 0 }[/math]
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