Template:Alta statistical properties summary: Difference between revisions

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(Created page with '===Statistical Properties Summary=== ====The Mean or MTTF==== The mean, <math>\overline{T},</math> or Mean Time To Failure (MTTF) of the 1-parameter exponential distribution is…')
 
 
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===Statistical Properties Summary===
#REDIRECT [[Distributions_Used_in_Accelerated_Testing]]
====The Mean or MTTF====
The mean,  <math>\overline{T},</math>  or Mean Time To Failure (MTTF) of the 1-parameter exponential distribution is given by:
<br>
::<math>\begin{align}
& \overline{T}= & \mathop{}_{0}^{\infty }t\cdot f(t)dt=\mathop{}_{0}^{\infty }t\cdot \lambda \cdot {{e}^{-\lambda t}}dt \\
& = & \frac{1}{\lambda } 
\end{align}</math>
 
====The Median====
The median, <math>\breve{T}</math>,
of the 1-parameter exponential distribution is given by:
<br>
::<math>\breve{T}=\frac{1}{\lambda }0.693</math>
<br>
====The Mode====
The mode,  <math>\tilde{T},</math>
of the 1-parameter exponential distribution is given by:
<br>
::<math>\tilde{T}=0</math>
<br>
 
====The Standard Deviation====
The standard deviation,  <math>{{\sigma }_{T}}</math> , of the 1-parameter exponential distribution is given by:
<br>
::<math>{{\sigma }_{T}}=\frac{1}{\lambda }=m</math>
 
<br>
====The Reliability Function====
The 1-parameter exponential reliability function is given by:
<br>
::<math>R(T)={{e}^{-\lambda T}}={{e}^{-\tfrac{T}{m}}}</math>
<br>
This function is the complement of the exponential cumulative distribution function or:
<br>
::<math>R(T)=1-Q(T)=1-\mathop{}_{0}^{T}f(T)dT</math>
<br>
:and:
<br>
::<math>R(T)=1-\mathop{}_{0}^{T}\lambda {{e}^{-\lambda T}}dT={{e}^{-\lambda T}}</math>
<br>
 
====Conditional Reliability====
<br>
The conditional reliability function for the 1-parameter exponential distribution is given by:
<br>
::<math>R(T,t)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\lambda t}}</math>
<br>
which says that the reliability for a mission of  <math>t</math>  duration undertaken after the component or equipment has already accumulated  <math>T</math>  hours of operation from age zero is only a function of the mission duration, and not a function of the age at the beginning of the mission. This is referred to as the ``memoryless property.''
<br>
====Reliable Life====
The reliable life, or the mission duration for a desired reliability goal,  <math>{{t}_{R}}</math> , for the 1-parameter exponential distribution is given by:
<br>
 
::<math>\begin{align}
  & R({{t}_{R}})= & {{e}^{-\lambda {{t}_{R}}}} \\
&  &  \\
& \ln [R({{t}_{R}})]= & -\lambda {{t}_{R}} 
\end{align}</math>
<br>
:or:
<br>
::<math>{{t}_{R}}=-\frac{\ln [R({{t}_{R}})]}{\lambda }</math>
 
<br>
====Failure Rate Function====
The exponential failure rate function is given by:
<br>
::<math>\lambda (T)=\frac{f(T)}{R(T)}=\frac{\lambda {{e}^{-\lambda (T)}}}{{{e}^{-\lambda (T)}}}=\lambda =\text{Constant}</math>
<br>

Latest revision as of 01:13, 16 August 2012

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