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| ===Arrhenius-Exponential Statistical Properties Summary===
| | #REDIRECT [[Arrhenius_Relationship#Arrhenius-Exponential]] |
| <br>
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| ====Mean or MTTF====
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| <br>
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| The mean, <math>\overline{T},</math> or Mean Time To Failure (MTTF) of the Arrhenius-exponential is given by,
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| <br>
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| ::<math>\begin{align}
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| & \overline{T}= & \mathop{}_{0}^{\infty }t\cdot f(t,V)dt=\mathop{}_{0}^{\infty }t\cdot \frac{1}{C{{e}^{\tfrac{B}{V}}}}{{e}^{-\tfrac{t}{C{{e}^{\tfrac{B}{V}}}}}}dt \\
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| & = & C{{e}^{\tfrac{B}{V}}}
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| \end{align}</math>
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| <br>
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| ====Median====
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| <br>
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| The median, <math>breve{T},</math>
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| of the Arrhenius-exponential model is given by:
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| ::<math>\breve{T}=0.693\cdot C{{e}^{\tfrac{B}{V}}}</math>
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| <br>
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| ====Mode====
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| The mode, <math>\tilde{T},</math>
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| of the Arrhenius-exponential model is given by:
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| ::<math>\tilde{T}=0</math>
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| <br>
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| ====Standard Deviation====
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| The standard deviation, <math>{{\sigma }_{T}}</math> , of the Arrhenius-exponential model is given by:
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| ::<math>{{\sigma }_{T}}=C{{e}^{\tfrac{B}{V}}}</math>
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| ====Arrhenius-Exponential Reliability Function====
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| The Arrhenius-exponential reliability function is given by:
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| <br>
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| ::<math>R(T,V)={{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}</math>
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| This function is the complement of the Arrhenius-exponential cumulative distribution function or:
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| ::<math>R(T,V)=1-Q(T,V)=1-\mathop{}_{0}^{T}f(T,V)dT</math>
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| and:
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| ::<math>R(T,V)=1-\mathop{}_{0}^{T}\frac{1}{C{{e}^{\tfrac{B}{V}}}}{{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}dT={{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}</math>
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| <br>
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| ====Conditional Reliability====
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| The Arrhenius-exponential conditional reliability function is given by,
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| ::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\tfrac{t}{C{{e}^{\tfrac{B}{V}}}}}}</math>
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| ====Reliable Life====
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| <br>
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| For the Arrhenius-exponential model, the reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}},</math> is given by:
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| ::<math>R({{t}_{R}},V)={{e}^{-\tfrac{{{t}_{R}}}{C{{e}^{\tfrac{B}{V}}}}}}</math>
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| <br>
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| ::<math>\ln [R({{t}_{R}},V)]=-\frac{{{t}_{R}}}{C{{e}^{\tfrac{B}{V}}}}</math>
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| or:
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| <br>
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| ::<math>{{t}_{R}}=-C{{e}^{\tfrac{B}{V}}}\ln [R({{t}_{R}},V)]</math>
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