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Arrhenius-Exponential Statistical Properties Summary

Mean or MTTF

The mean, [math]\displaystyle{ \overline{T}, }[/math] or Mean Time To Failure (MTTF) of the Arrhenius-exponential is given by,

[math]\displaystyle{ \begin{align} & \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 \\ & = & C{{e}^{\tfrac{B}{V}}} \end{align} }[/math]

Median

The median, [math]\displaystyle{ breve{T}, }[/math] of the Arrhenius-exponential model is given by:

[math]\displaystyle{ \breve{T}=0.693\cdot C{{e}^{\tfrac{B}{V}}} }[/math]

Mode

The mode, [math]\displaystyle{ \tilde{T}, }[/math] of the Arrhenius-exponential model is given by:

[math]\displaystyle{ \tilde{T}=0 }[/math]

Standard Deviation

The standard deviation, [math]\displaystyle{ {{\sigma }_{T}} }[/math] , of the Arrhenius-exponential model is given by:

[math]\displaystyle{ {{\sigma }_{T}}=C{{e}^{\tfrac{B}{V}}} }[/math]

Arrhenius-Exponential Reliability Function

The Arrhenius-exponential reliability function is given by:

[math]\displaystyle{ R(T,V)={{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}} }[/math]

This function is the complement of the Arrhenius-exponential cumulative distribution function or:

[math]\displaystyle{ R(T,V)=1-Q(T,V)=1-\mathop{}_{0}^{T}f(T,V)dT }[/math]

and:

[math]\displaystyle{ 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]

Conditional Reliability

The Arrhenius-exponential conditional reliability function is given by,

[math]\displaystyle{ 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

For the Arrhenius-exponential model, the reliable life, or the mission duration for a desired reliability goal, [math]\displaystyle{ {{t}_{R}}, }[/math] is given by:

[math]\displaystyle{ R({{t}_{R}},V)={{e}^{-\tfrac{{{t}_{R}}}{C{{e}^{\tfrac{B}{V}}}}}} }[/math]

[math]\displaystyle{ \ln [R({{t}_{R}},V)]=-\frac{{{t}_{R}}}{C{{e}^{\tfrac{B}{V}}}} }[/math]

or:

[math]\displaystyle{ {{t}_{R}}=-C{{e}^{\tfrac{B}{V}}}\ln [R({{t}_{R}},V)] }[/math]