Template:Lognormal distribution probability density function

Lognormal Probability Density Function

The lognormal distribution is a two-parameter distribution with parameters [math]\displaystyle{ {\mu }' }[/math] and [math]\displaystyle{ {{\sigma }'}}} }[/math] . The [math]\displaystyle{ pdf }[/math] for this distribution is given by:

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

where, [math]\displaystyle{ {T}'=\ln (T) }[/math]. , where the [math]\displaystyle{ T }[/math] values are the times-to-failure, and

[math]\displaystyle{ \mu'=\text{mean of the natural logarithms} }[/math]

[math]\displaystyle{ \text{of the times-to-failure,} }[/math]

[math]\displaystyle{ \sigma_{T'}=\text{standard deviation of the natural logarithms} }[/math]

[math]\displaystyle{ \text{of the times-to-failure} }[/math]

The lognormal [math]\displaystyle{ pdf }[/math] can be obtained, realizing that for equal probabilities under the normal and lognormal [math]\displaystyle{ pdf }[/math] s, incremental areas should also be equal, or:

[math]\displaystyle{ f(T)dT=f({T}')d{T}' }[/math]

Taking the derivative yields:

[math]\displaystyle{ d{T}'=\frac{dT}{T} }[/math]

Substitution yields:

[math]\displaystyle{ \begin{align} f(T)= & \frac{f({T}')}{T}, \\ f(T)= & \frac{1}{T\cdot {{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(T)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}} \end{align} }[/math]

where:

[math]\displaystyle{ f(T)\ge 0,T\gt 0,-\infty \lt {\mu }'\lt \infty ,{{\sigma }_{{{T}'}}}\gt 0 }[/math]