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| ===Statistical Properties Summary===
| | #REDIRECT [[Distributions_Used_in_Accelerated_Testing]] |
| {{alta exponential mean}}
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| {{alta exponential median}}
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| {{alta exponential mode}}
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| ====The Standard Deviation====
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| The standard deviation, <math>{{\sigma }_{T}}</math> , of the 1-parameter exponential distribution is given by:
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| <br>
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| ::<math>{{\sigma }_{T}}=\frac{1}{\lambda }=m</math>
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| <br>
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| ====The Reliability Function====
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| The 1-parameter exponential reliability function is given by:
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| <br>
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| ::<math>R(T)={{e}^{-\lambda T}}={{e}^{-\tfrac{T}{m}}}</math>
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| <br>
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| This function is the complement of the exponential cumulative distribution function or:
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| <br>
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| ::<math>R(T)=1-Q(T)=1-\mathop{}_{0}^{T}f(T)dT</math>
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| <br>
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| :and:
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| <br>
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| ::<math>R(T)=1-\mathop{}_{0}^{T}\lambda {{e}^{-\lambda T}}dT={{e}^{-\lambda T}}</math>
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| <br>
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| ====Conditional Reliability====
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| <br>
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| The conditional reliability function for the 1-parameter exponential distribution is given by:
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| <br>
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| ::<math>R(T,t)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\lambda t}}</math>
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| <br>
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| 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.''
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| <br>
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| ====Reliable Life====
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| 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:
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| <br>
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| ::<math>\begin{align}
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| & R({{t}_{R}})= & {{e}^{-\lambda {{t}_{R}}}} \\
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| & & \\
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| & \ln [R({{t}_{R}})]= & -\lambda {{t}_{R}}
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| \end{align}</math>
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| <br>
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| :or:
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| <br>
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| ::<math>{{t}_{R}}=-\frac{\ln [R({{t}_{R}})]}{\lambda }</math>
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| <br>
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| ====Failure Rate Function====
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| The exponential failure rate function is given by:
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| <br>
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| ::<math>\lambda (T)=\frac{f(T)}{R(T)}=\frac{\lambda {{e}^{-\lambda (T)}}}{{{e}^{-\lambda (T)}}}=\lambda =\text{Constant}</math>
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| <br>
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