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===Arrhenius-Exponential Statistical Properties Summary===
#REDIRECT [[Arrhenius_Relationship#Arrhenius-Exponential]]
<br>
{{aae mean}}
====Median====
<br>
The median, <math>breve{T},</math>
of the Arrhenius-exponential model is given by:
 
<br>
::<math>\breve{T}=0.693\cdot C{{e}^{\tfrac{B}{V}}}</math>
 
 
<br>
 
====Mode====
<br>
The mode,  <math>\tilde{T},</math> 
of the Arrhenius-exponential model is given by:
 
<br>
::<math>\tilde{T}=0</math>
 
<br>
 
====Standard Deviation====
The standard deviation,  <math>{{\sigma }_{T}}</math> , of the Arrhenius-exponential model is given by:
 
<br>
::<math>{{\sigma }_{T}}=C{{e}^{\tfrac{B}{V}}}</math>
 
====Arrhenius-Exponential Reliability Function====
<br>
 
The Arrhenius-exponential reliability function is given by:
 
<br>
::<math>R(T,V)={{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}</math>
 
<br>
This function is the complement of the Arrhenius-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}\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>
 
<br>
 
====Conditional Reliability====
The Arrhenius-exponential conditional reliability function is given by,
 
::<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>
 
====Reliable Life====
<br>
For the Arrhenius-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}^{-\tfrac{{{t}_{R}}}{C{{e}^{\tfrac{B}{V}}}}}}</math>
 
 
 
<br>
::<math>\ln [R({{t}_{R}},V)]=-\frac{{{t}_{R}}}{C{{e}^{\tfrac{B}{V}}}}</math>
 
<br>
or:
 
<br>
::<math>{{t}_{R}}=-C{{e}^{\tfrac{B}{V}}}\ln [R({{t}_{R}},V)]</math>

Latest revision as of 00:06, 16 August 2012

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