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:


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

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


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


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


 * $$\sigma'=\text{standard deviation of the natural logarithms}$$


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

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 \sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(t)-{\mu }'}{} \right)}^{2}}}} \end{align}$$


 * where:


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