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==Normal Statistical Properties==
#REDIRECT [[The_Normal_Distribution]]
 
{{normal mean median and mode}}
 
===The Normal Standard Deviation===
 
As with the mean, the standard deviation for the normal distribution is actually one of the parameters, usually denoted as  <math>{{\sigma }_{T}}.</math>
 
===The Normal Reliability Function===
The reliability for a mission of time  <math>T</math>  for the normal distribution is determined by:
 
::<math>R(T)=\int_{T}^{\infty }f(t)dt=\int_{T}^{\infty }\frac{1}{{{\sigma }_{T}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\mu }{{{\sigma }_{T}}} \right)}^{2}}}}dt</math>
 
There is no closed-form solution for the normal 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 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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