Template:Lognormal distribution statistical properties: Difference between revisions

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(Created page with '==Lognormal Statistical Properties== ===The Mean or MTTF=== The mean of the lognormal distribution, <math>\mu </math> , is given by [18]: ::<math>\mu ={{e}^{{\mu }'+\tfrac{1}{…')
 
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==Lognormal Statistical Properties==
#REDIRECT [[Lognormal Distribution Functions]]
===The Mean or MTTF===
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.
 
===The Lognormal Conditional Reliability===
The lognormal conditional reliability function is given by:
 
::<math>R(t|T)=\frac{R(T+t)}{R(T)}=\frac{\int_{\text{ln}(T+t)}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{s-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}ds}{\int_{\text{ln}(T)}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{s-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}ds}</math>
 
Once again, the use of standard normal tables is necessary to solve this equation, as no closed-form solution exists.
 
===The Lognormal Reliable Life===
As there is no closed-form solution for the lognormal reliability equation, no closed-form solution exists for the lognormal reliable life either.  In order to determine this value, one must solve the equation:
 
 
::<math>{{R}_{T}}=\int_{\text{ln}(T)}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{s-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}ds</math>
 
for <math>T</math> .
 
===The Lognormal Failure Rate Function===
The lognormal failure rate is given by:
 
 
::<math>\lambda (T)=\frac{f(T)}{R(T)}=\frac{\tfrac{1}{T\cdot {{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{{T}'-{\mu }'}{{{\sigma }_{{{T}'}}}})}^{2}}}}}{\int_{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{t-{\mu }'}{{{\sigma }_{{{T}'}}}})}^{2}}}}dt}</math>
 
As with the reliability equations, standard normal tables will be required to solve for this function.

Latest revision as of 04:44, 13 August 2012

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