Template:Normal statistical properties: Difference between revisions

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==Normal Statistical Properties==
#REDIRECT [[The_Normal_Distribution]]
 
{{normal mean median and mode}}
 
{{normal standard deviation}}
 
{{normal reliability function}}
 
===The Normal Conditional Reliability Function===
 
The normal conditional reliability function is given by:
 
::<math>R(t|T)=\frac{R(T+t)}{R(T)}=\frac{\int_{T+t}^{\infty }\tfrac{1}{{{\sigma }_{T}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\mu }{{{\sigma }_{T}}} \right)}^{2}}}}dt}{\int_{T}^{\infty }\tfrac{1}{{{\sigma }_{T}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\mu }{{{\sigma }_{T}}} \right)}^{2}}}}dt}</math>
 
Once again, the use of standard normal tables for the calculation of the normal conditional reliability is necessary, as there is no closed form solution.
 
 
===The Normal Reliable Life===
 
Since there is no closed-form solution for the normal reliability function, there will also be no closed-form solution for the normal reliable life. To determine the normal reliable life, one must solve:
 
::<math>R(T)=\int_{T}^{\infty }\frac{1}{{{\sigma }_{T}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\mu }{{{\sigma }_{T}}} \right)}^{2}}}}dt</math>
::for  <math>T</math> .
 
===The Normal Failure Rate Function===
 
The instantaneous normal failure rate is given by:
 
::<math>\lambda (T)=\frac{f(T)}{R(T)}=\frac{\tfrac{1}{{{\sigma }_{T}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{T-\mu }{{{\sigma }_{T}}} \right)}^{2}}}}}{\int_{T}^{\infty }\tfrac{1}{{{\sigma }_{T}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\mu }{{{\sigma }_{T}}} \right)}^{2}}}}dt}</math>

Latest revision as of 02:48, 13 August 2012

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