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| ===Eyring-Exponential Statistical Properties Summary===
| | #REDIRECT [[Eyring_Relationship#Eyring-Exponential]] |
| ====Mean or MTTF====
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| <br>
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| The mean, <math>\overline{T},</math> or Mean Time To Failure (MTTF) for the Eyring-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 V{{e}^{\left( A-\tfrac{B}{V} \right)}}{{e}^{-tV{{e}^{\left( A-\tfrac{B}{V} \right)}}}}dt \\
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| & = & \frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}
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| \end{align}</math>
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|
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| ====Median====
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| <br>
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| The median, <math>\breve{T},</math>
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| for the Eyring-exponential model is given by:
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| <br>
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| ::<math>\breve{T}=0.693\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>
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| <br>
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| ====Mode====
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| <br>
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| The mode, <math>\tilde{T},</math>
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| for the Eyring-exponential model is <math>\tilde{T}=0.</math>
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| <br>
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| ====Standard Deviation====
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| <br>
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| The standard deviation, <math>{{\sigma }_{T}}</math>, for the Eyring-exponential model is given by:
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| <br>
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| ::<math>{{\sigma }_{T}}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>
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| <br>
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| ====Eyring-Exponential Reliability Function====
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| <br>
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| The Eyring-exponential reliability function is given by:
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| <br>
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| ::<math>R(T,V)={{e}^{-T\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>
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| <br>
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| This function is the complement of the Eyring-exponential cumulative distribution function or:
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| <br>
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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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| <br>
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| :and:
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| <br>
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| ::<math>R(T,V)=1-\mathop{}_{0}^{T}V{{e}^{\left( A-\tfrac{B}{V} \right)}}{{e}^{-T\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}dT={{e}^{-T\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>
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| ====Conditional Reliability====
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| <br>
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| The conditional reliability function for the Eyring-exponential model is given by:
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| <br>
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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}^{-t\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>
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| <br>
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| ====Reliable Life====
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| <br>
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| For the Eyring-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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| <br>
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| ::<math>R({{t}_{R}},V)={{e}^{-{{t}_{R}}\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>
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| <br>
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| ::<math>\ln [R({{t}_{R}},V)]=-{{t}_{R}}\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}</math>
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| <br>
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| :or:
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| <br>
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| ::<math>{{t}_{R}}=-\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}\ln [R({{t}_{R}},V)]</math>
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