Template:Alta statistical properties summary

Statistical Properties Summary

The Mean or MTTF

The mean, [math]\displaystyle{ \overline{T}, }[/math] or Mean Time To Failure (MTTF) of the 1-parameter exponential distribution is given by:

[math]\displaystyle{ \begin{align} & \overline{T}= & \mathop{}_{0}^{\infty }t\cdot f(t)dt=\mathop{}_{0}^{\infty }t\cdot \lambda \cdot {{e}^{-\lambda t}}dt \\ & = & \frac{1}{\lambda } \end{align} }[/math]

The Median

The median, [math]\displaystyle{ \breve{T} }[/math], of the 1-parameter exponential distribution is given by:

[math]\displaystyle{ \breve{T}=\frac{1}{\lambda }0.693 }[/math]

The Mode

The mode, [math]\displaystyle{ \tilde{T}, }[/math] of the 1-parameter exponential distribution is given by:

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

The Standard Deviation

The standard deviation, [math]\displaystyle{ {{\sigma }_{T}} }[/math] , of the 1-parameter exponential distribution is given by:

[math]\displaystyle{ {{\sigma }_{T}}=\frac{1}{\lambda }=m }[/math]

The Reliability Function

The 1-parameter exponential reliability function is given by:

[math]\displaystyle{ R(T)={{e}^{-\lambda T}}={{e}^{-\tfrac{T}{m}}} }[/math]

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

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

and:

[math]\displaystyle{ R(T)=1-\mathop{}_{0}^{T}\lambda {{e}^{-\lambda T}}dT={{e}^{-\lambda T}} }[/math]

Conditional Reliability

The conditional reliability function for the 1-parameter exponential distribution is given by:

[math]\displaystyle{ R(T,t)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\lambda t}} }[/math]

which says that the reliability for a mission of [math]\displaystyle{ t }[/math] duration undertaken after the component or equipment has already accumulated [math]\displaystyle{ T }[/math] hours of operation from age zero is only a function of the mission duration, and not a function of the age at the beginning of the mission. This is referred to as the ``memoryless property.

Reliable Life

The reliable life, or the mission duration for a desired reliability goal, [math]\displaystyle{ {{t}_{R}} }[/math] , for the 1-parameter exponential distribution is given by:

[math]\displaystyle{ \begin{align} & R({{t}_{R}})= & {{e}^{-\lambda {{t}_{R}}}} \\ & & \\ & \ln [R({{t}_{R}})]= & -\lambda {{t}_{R}} \end{align} }[/math]

or:

[math]\displaystyle{ {{t}_{R}}=-\frac{\ln [R({{t}_{R}})]}{\lambda } }[/math]

Failure Rate Function

The exponential failure rate function is given by:

[math]\displaystyle{ \lambda (T)=\frac{f(T)}{R(T)}=\frac{\lambda {{e}^{-\lambda (T)}}}{{{e}^{-\lambda (T)}}}=\lambda =\text{Constant} }[/math]