Template:Eyring-log stat prop sum

Eyring-Lognormal Statistical Properties Summary

The Mean

• The mean life of the Eyring-lognormal model (mean of the times-to-failure), [math]\displaystyle{ \bar{T} }[/math] , is given by:

[math]\displaystyle{ \begin{align} & \bar{T}= & {{e}^{\bar{{T}'}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}} \\ & = & {{e}^{-\ln (V)-A+\tfrac{B}{V}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}} \end{align} }[/math]

The mean of the natural logarithms of the times-to-failure, [math]\displaystyle{ {{\bar{T}}^{^{\prime }}} }[/math] , in terms of [math]\displaystyle{ \bar{T} }[/math] and [math]\displaystyle{ {{\sigma }_{T}} }[/math] is given by:

[math]\displaystyle{ {{\bar{T}}^{\prime }}=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right) }[/math]

The Median

The median of the Eyring-lognormal model is given by:

[math]\displaystyle{ \breve{T}={{e}^{{{\overline{T}}^{\prime }}}} }[/math]

The Standard Deviation

• The standard deviation of the Eyring-lognormal model (standard deviation of the times-to-failure), [math]\displaystyle{ {{\sigma }_{T}} }[/math] , is given by:

[math]\displaystyle{ \begin{align} & {{\sigma }_{T}}= & \sqrt{\left( {{e}^{2\bar{{T}'}+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} \\ & = & \sqrt{\left( {{e}^{2\left( -\ln (V)-A+\tfrac{B}{V} \right)+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} \end{align} }[/math]

• The standard deviation of the natural logarithms of the times-to-failure, [math]\displaystyle{ {{\sigma }_{{{T}'}}} }[/math] , in terms of [math]\displaystyle{ \bar{T} }[/math] and [math]\displaystyle{ {{\sigma }_{T}} }[/math] is given by:

[math]\displaystyle{ {{\sigma }_{{{T}'}}}=\sqrt{\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)} }[/math]

The Mode

• The mode of the Eyring-lognormal model is given by:

[math]\displaystyle{ \begin{align} & \tilde{T}= & {{e}^{{{\overline{T}}^{\prime }}-\sigma _{{{T}'}}^{2}}} \\ & = & {{e}^{-\ln (V)-A+\tfrac{B}{V}-\sigma _{{{T}'}}^{2}}} \end{align} }[/math]

Eyring-Lognormal Reliability Function

The reliability for a mission of time [math]\displaystyle{ T }[/math] , starting at age 0, for the Eyring-lognormal model is determined by:

[math]\displaystyle{ R(T,V)=\mathop{}_{T}^{\infty }f(t,V)dt }[/math]

or:

[math]\displaystyle{ R(T,V)=\mathop{}_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt }[/math]

There is no closed form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables. Since the application automatically solves for the reliability we will not discuss manual solution methods.

Reliable Life

For the Eyring-lognormal model, the reliable life, or the mission duration for a desired reliability goal, [math]\displaystyle{ {{t}_{R}}, }[/math] is estimated by first solving the reliability equation with respect to time, as follows:

[math]\displaystyle{ T_{R}^{\prime }=-\ln (V)-A+\frac{B}{V}+z\cdot {{\sigma }_{{{T}'}}} }[/math]

where:

[math]\displaystyle{ z={{\Phi }^{-1}}\left[ F\left( T_{R}^{\prime },V \right) \right] }[/math]

and:

[math]\displaystyle{ \Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}',V)}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt }[/math]

Since [math]\displaystyle{ {T}'=\ln (T) }[/math] the reliable life, [math]\displaystyle{ {{t}_{R,}} }[/math] is given by:

[math]\displaystyle{ {{t}_{R}}={{e}^{T_{R}^{\prime }}} }[/math]

Eyring-Lognormal Failure Rate

The Eyring-lognormal failure rate is given by:

[math]\displaystyle{ \lambda (T,V)=\frac{f(T,V)}{R(T,V)}=\frac{\tfrac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}}{\mathop{}_{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt} }[/math]