Template:Eyring-ex stat prop sum

Eyring-Exponential Statistical Properties Summary

Mean or MTTF

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

[math]\displaystyle{ \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

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

[math]\displaystyle{ \breve{T}=0.693\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}} }[/math]

Mode

The mode, [math]\displaystyle{ \tilde{T}, }[/math] for the Eyring-exponential model is [math]\displaystyle{ \tilde{T}=0. }[/math]

Standard Deviation

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

[math]\displaystyle{ {{\sigma }_{T}}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}} }[/math]

Eyring-Exponential Reliability Function

The Eyring-exponential reliability function is given by:

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

This function is the complement of the Eyring-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}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

The conditional reliability function for the Eyring-exponential model 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}^{-t\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}} }[/math]

Reliable Life

For the Eyring-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}^{-{{t}_{R}}\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}} }[/math]

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

or:

[math]\displaystyle{ {{t}_{R}}=-\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}\ln [R({{t}_{R}},V)] }[/math]