Template:Aw statistical properties summary

Statistical Properties Summary

The Mean or MTTF

The mean, [math]\displaystyle{ \overline{T}, }[/math] of the 2-parameter Weibull [math]\displaystyle{ pdf }[/math] is given by:

[math]\displaystyle{ \overline{T}=\eta \cdot \Gamma \left( \frac{1}{\beta }+1 \right) }[/math]

where [math]\displaystyle{ \Gamma \left( \tfrac{1}{\beta }+1 \right) }[/math] is the gamma function evaluated at the value of [math]\displaystyle{ \left( \tfrac{1}{\beta }+1 \right) }[/math] .

The Median

The median, [math]\displaystyle{ \breve{T}, }[/math] of the 2-parameter Weibull is given by:

[math]\displaystyle{ \breve{T}=\eta {{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}} }[/math]

The Mode

The mode, [math]\displaystyle{ \tilde{T}, }[/math] of the 2-parameter Weibull is given by:

[math]\displaystyle{ \tilde{T}=\eta {{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }}} }[/math]

The Standard Deviation

The standard deviation, [math]\displaystyle{ {{\sigma }_{T}}, }[/math] of the 2-parameter Weibull is given by:

[math]\displaystyle{ {{\sigma }_{T}}=\eta \cdot \sqrt{\Gamma \left( \frac{2}{\beta }+1 \right)-\Gamma {{\left( \frac{1}{\beta }+1 \right)}^{2}}} }[/math]

The [math]\displaystyle{ cdf }[/math] and the Reliability Function

The [math]\displaystyle{ cdf }[/math] of the 2-parameter Weibull distribution is given by:

[math]\displaystyle{ F(T)=1-{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}} }[/math]

The Weibull reliability function is given by:

[math]\displaystyle{ \begin{align} & R(T)= & 1-F(t) \\ & = & {{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}} \end{align} }[/math]

The Conditional Reliability Function

The Weibull conditional reliability function is given by:

[math]\displaystyle{ R(T,t)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}} }[/math]

or:

[math]\displaystyle{ R(T,t)={{e}^{-\left[ {{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}-{{\left( \tfrac{T}{\eta } \right)}^{\beta }} \right]}} }[/math]

Eqn. (e137) gives the reliability for a new mission of [math]\displaystyle{ t }[/math] duration, having already accumulated [math]\displaystyle{ T }[/math] hours of operation up to the start of this new mission, and the units are checked out to assure that they will start the next mission successfully. (It is called conditional because you can calculate the reliability of a new mission based on the fact that the unit(s) already accumulated [math]\displaystyle{ T }[/math] hours of operation successfully.)

The Reliable Life

For the 2-parameter Weibull distribution, the reliable life, [math]\displaystyle{ {{T}_{R}} }[/math] , of a unit for a specified reliability, starting the mission at age zero, is given by:

[math]\displaystyle{ {{T}_{R}}=\eta \cdot {{\left\{ -\ln \left[ R\left( {{T}_{R}} \right) \right] \right\}}^{\tfrac{1}{\beta }}} }[/math]

This is the life for which the unit will function successfully with a reliability of [math]\displaystyle{ R({{T}_{R}}) }[/math] . If [math]\displaystyle{ R({{T}_{R}})=0.50 }[/math] then [math]\displaystyle{ {{T}_{R}}=\breve{T} }[/math], the median life, or the life by which half of the units will survive.

The Failure Rate Function

The 2-parameter Weibull failure rate function,[math]\displaystyle{ \lambda (T) }[/math] , is given by:

[math]\displaystyle{ \lambda \left( T \right)=\frac{f\left( T \right)}{R\left( T \right)}=\frac{\beta }{\eta }{{\left( \frac{T}{\eta } \right)}^{\beta -1}} }[/math]