Lognormal Statistical Properties

The Mean or MTTF

The mean of the lognormal distribution, [math]\displaystyle{ \mu }[/math] , is given by [18]:

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

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

[math]\displaystyle{ {\mu }'=\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 lognormal distribution, [math]\displaystyle{ \breve{T} }[/math] , is given by [18]:

[math]\displaystyle{ \breve{T}={{e}^{{{\mu }'}}} }[/math]

The Mode

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

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

The Standard Deviation

The standard deviation of the lognormal distribution, [math]\displaystyle{ {{\sigma }_{T}} }[/math] , is given by [18]:

[math]\displaystyle{ {{\sigma }_{T}}=\sqrt{\left( {{e}^{2{\mu }'+\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 Lognormal Reliability Function

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

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

or:

[math]\displaystyle{ R(T)=\int_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt }[/math]

As with the normal distribution, 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. For interested readers, full explanations can be found in the references.