The Lognormal Distribution

The lognormal distribution is commonly used to model the lives of units whose failure modes are of a fatigue-stress nature. Since this includes most, if not all, mechanical systems, the lognormal distribution can have widespread application. Consequently, the lognormal distribution is a good companion to the Weibull distribution when attempting to model these types of units. As may be surmised by the name, the lognormal distribution has certain similarities to the normal distribution. A random variable is lognormally distributed if the logarithm of the random variable is normally distributed. Because of this, there are many mathematical similarities between the two distributions. For example, the mathematical reasoning for the construction of the probability plotting scales and the bias of parameter estimators is very similar for these two distributions.

Lognormal Probability Density Function
The lognormal distribution is a 2-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:


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


 * $$\mu'\,\!$$ = mean of the natural logarithms of the times-to-failure


 * $$\sigma'\,\!$$ = standard deviation of the natural logarithms of the times-to-failure

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


 * $$\begin{align}

f(t)dt=f({t}')d{t}' \end{align}\,\!$$

Taking the derivative of the relationship between $${t}'\,\!$$ and $${t}\,\!$$ 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:


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