Template:Lognormal distribution probability density function

Lognormal Probability Density Function
The lognormal distribution is a two-parameter distribution with parameters $${\mu }'$$  and  $${{\sigma }_}$$. The $$pdf$$  for this distribution is given by:


 * $$f({T}')=\frac{1}{{{\sigma }_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{{T}^{\prime }}-{\mu }'}{{{\sigma }_}} \right)}^{2}}}}$$

where, $${T}'=\ln (T)$$. , where the $$T$$  values are the times-to-failure, and


 * $$\mu'=\text{mean of the natural logarithms}$$


 * $$\text{of the times-to-failure,}$$


 * $$\sigma_{T'}=\text{standard deviation of the natural logarithms}$$


 * $$\text{of the times-to-failure}$$

The lognormal $$pdf$$  can be obtained, realizing that for equal probabilities under the normal and lognormal  $$pdf$$ s, incremental areas should also be equal, or:


 * $$f(T)dT=f({T}')d{T}'$$

Taking the derivative yields:


 * $$d{T}'=\frac{dT}{T}$$


 * Substitution yields:


 * $$\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}$$


 * where:


 * $$f(T)\ge 0,T>0,-\infty <{\mu }'<\infty ,{{\sigma }_}>0$$