Template:Lognormal distribution statistical properties: Difference between revisions

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==Lognormal Statistical Properties==
#REDIRECT [[Lognormal Distribution Functions]]
{{ld mean}}
 
{{ld median}}
 
{{ld mode}}
 
{{ld standard deviation}}
 
{{ld lognormal reliability function}}
 
===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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