Weibull++ Standard Folio Data Lognormal: Difference between revisions

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The lognormal distribution is a two-parameter distribution with parameters  <br>  
The lognormal distribution is a two-parameter distribution with parameters  <br>  
<math>{\mu }'</math>  and  <math>{{\sigma }_{{{T}'}}}</math>. <br> The ''pdf'' is given by:  
<math>{\mu }'</math>  and  <math>{{\sigma }_{{{T}'}}}</math>. <br> The ''pdf'' 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>
::<math>f({T}')=\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{{T}^{\prime }}-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
<br> where,  
<br> where,  
<br><math>{T}'=\ln (T)</math>. , where the  <math>T</math>  values are the times-to-failure, and
<br><math>{T}'=\ln (T)</math> and
:<math>\mu'=\text{mean of the natural logarithms}</math>
<br><math>\mu' \text{ and } <math>\sigma_{T'}</math>\
are the mean and standard deviation of of the natural logarithms of the times-to-failure.
:<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>
 
 
 
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<br>
 




Revision as of 19:06, 11 November 2011

Reliability Web Notes

Weibull Folio
Life Data Analysis

The lognormal distribution is commonly used to model the lives of units whose failure modes are of a fatigue-stress nature. It has an increasing failure rate behavior and then decreasing towards the end of life.

The lognormal distribution is a two-parameter distribution with parameters
[math]\displaystyle{ {\mu }' }[/math] and [math]\displaystyle{ {{\sigma }_{{{T}'}}} }[/math].
The pdf is given by:

[math]\displaystyle{ 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]\displaystyle{ {T}'=\ln (T) }[/math] and
[math]\displaystyle{ \mu' \text{ and } \lt math\gt \sigma_{T'} }[/math]\ are the mean and standard deviation of of the natural logarithms of the times-to-failure.

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