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' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{{t}^{\prime }}-{\mu }'}{{{\sigma' }}} \right)}^{2}}}} }[/math]

where,

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

[math]\displaystyle{ \mu'=\text{mean of the natural logarithms of the times-to-failure,} }[/math]

[math]\displaystyle{ \sigma'=\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' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(t)-{\mu }'}{{{\sigma' }}} \right)}^{2}}}} \end{align} }[/math]

where:

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