Template:Alta ld statistical prop func

Statistical Properties Summary

The Mean or MTTF

• The mean of the lognormal distribution, [math]\displaystyle{ \bar{T} }[/math] , is given by:

[math]\displaystyle{ \bar{T}={{e}^{\bar{{T}'}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}} }[/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 Standard Deviation

• The standard deviation of the lognormal distribution, [math]\displaystyle{ {{\sigma }_{T}} }[/math] , is given by:

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

• The median of the lognormal distribution is given by:

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

The Mode

• The mode of the lognormal distribution is given by:

[math]\displaystyle{ \tilde{T}={{e}^{{{\bar{T}}^{\prime }}-\sigma _{{{T}'}}^{2}}} }[/math]

Reliability Function

For the lognormal distribution, the reliability for a mission of time [math]\displaystyle{ T }[/math] , starting at age 0, is given by:

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

or:

[math]\displaystyle{ R(T)=\mathop{}_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\overline{{{T}'}}}{{{\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.

Lognormal Failure Rate

The lognormal failure rate is given by:

[math]\displaystyle{ \lambda (T)=\frac{f(T)}{R(T)}=\frac{\tfrac{1}{{T}'{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}})}^{2}}}}}{\mathop{}_{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{t-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}})}^{2}}}}dt} }[/math]