Lognormal Statistical Properties: Difference between revisions

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====The Mean or MTTF====
#REDIRECT [[Lognormal Distribution Functions]]
The mean of the lognormal distribution,  <math>\mu </math> , is given by [18]:
 
::<math>\mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}}</math>
 
 
The mean of the natural logarithms of the times-to-failure, <math>\mu'</math> , in terms of  <math>\bar{T}</math>  and  <math>{{\sigma }_{T}}</math>  is givgen by:
 
::<math>{\mu }'=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)</math>
 
====The Median====
The median of the lognormal distribution,  <math>\breve{T}</math> , is given by [18]:
 
::<math>\breve{T}={{e}^{{{\mu }'}}}</math>
 
====The Mode====
The mode of the lognormal distribution,  <math>\tilde{T}</math> , is given by [1]:
 
::<math>\tilde{T}={{e}^{{\mu }'-\sigma _{{{T}'}}^{2}}}</math>
 
====The Standard Deviation====
The standard deviation of the lognormal distribution,  <math>{{\sigma }_{T}}</math> , is given by [18]:
 
::<math>{{\sigma }_{T}}=\sqrt{\left( {{e}^{2{\mu }'+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)}</math>
 
 
The standard deviation of the natural logarithms of the times-to-failure,  <math>{{\sigma }_{{{T}'}}}</math> , in terms of  <math>\bar{T}</math>  and  <math>{{\sigma }_{T}}</math>  is given by:
 
::<math>{{\sigma }_{{{T}'}}}=\sqrt{\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)}</math>
 
 
====The Lognormal Reliability Function====
The reliability for a mission of time  <math>T</math> , starting at age 0, for the lognormal distribution is determined by:
 
::<math>R(T)=\int_{T}^{\infty }f(t)dt</math>
 
or:
 
::<math>R(T)=\int_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt</math>
 
As with the normal distribution, there is no closed-form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables. Since the application automatically solves for the reliability we will not discuss manual solution methods. For interested readers, full explanations can be found in the references.

Latest revision as of 05:05, 13 August 2012

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