Template:Eyring-ex stat prop sum: Difference between revisions

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(Created page with '===Eyring-Exponential Statistical Properties Summary=== ====Mean or MTTF==== <br> The mean, <math>\overline{T},</math> or Mean Time To Failure (MTTF) for the Eyring-exponential i…')
 
 
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===Eyring-Exponential Statistical Properties Summary===
#REDIRECT [[Eyring_Relationship#Eyring-Exponential]]
====Mean or MTTF====
<br>
The mean, <math>\overline{T},</math> or Mean Time To Failure (MTTF) for the Eyring-exponential is given by:
 
<br>
::<math>\begin{align}
  & \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 \\
& = & \frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}} 
\end{align}</math>
====Median====
<br>
 
The median, <math>\breve{T},</math>
for the Eyring-exponential model is given by:
 
<br>
 
::<math>\breve{T}=0.693\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>
 
<br>
 
====Mode====
<br>
The mode,  <math>\tilde{T},</math>
for the Eyring-exponential model is <math>\tilde{T}=0.</math>
<br>
====Standard Deviation====
<br>
The standard deviation,  <math>{{\sigma }_{T}}</math>, for the Eyring-exponential model is given by:
<br>
::<math>{{\sigma }_{T}}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>
<br>
====Eyring-Exponential Reliability Function====
<br>
The Eyring-exponential reliability function is given by:
 
<br>
::<math>R(T,V)={{e}^{-T\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>
 
<br>
This function is the complement of the Eyring-exponential cumulative distribution function or:
 
<br>
::<math>R(T,V)=1-Q(T,V)=1-\mathop{}_{0}^{T}f(T,V)dT</math>
 
<br>
:and:
 
<br>
::<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>
 
====Conditional Reliability====
<br>
The conditional reliability function for the Eyring-exponential model is given by:
<br>
::<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>
<br>
====Reliable Life====
<br>
For the Eyring-exponential model, the reliable life, or the mission duration for a desired reliability goal,  <math>{{t}_{R,}}</math>  is given by:
<br>
::<math>R({{t}_{R}},V)={{e}^{-{{t}_{R}}\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>
<br>
::<math>\ln [R({{t}_{R}},V)]=-{{t}_{R}}\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}</math>
<br>
:or:
<br>
::<math>{{t}_{R}}=-\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}\ln [R({{t}_{R}},V)]</math>

Latest revision as of 22:59, 16 August 2012

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