21.4.7.2. Lognormal Distribution

In many engineering problems, a random variable should be positive value due to the physical limitation. Thus, a lognormal distribution is widely used for the random variables, which is a normal distribution of \(Y=\ln (X)\) for the random variables (\(X\)). The PDF of lognormal distribution can be expressed as

\({{f}_{Y}}(y)=\frac{1}{{{\sigma }_{Y}}\sqrt{2\pi }}\exp \left[ -\frac{1}{2}{{\left( \frac{y-{{\mu }_{Y}}}{{{\sigma }_{Y}}} \right)}^{2}} \right],-\propto \le y\le +\propto\)

Now, in terms of x, the above PDF can be expressed as

\({{f}_{X}}(x)=\frac{1}{{{\sigma }_{Y}}\sqrt{2\pi }}\exp \left[ -\frac{1}{2}{{\left( \frac{\ln (x)-{{\mu }_{Y}}}{{{\sigma }_{Y}}} \right)}^{2}} \right],0\le x\)

where \(\sigma _{Y}^{2}=\ln \left[ {{\left( \frac{\sigma X}{\mu X} \right)}^{2}}+1 \right]\) and \({{\mu }_{Y}}=\ln ({{\mu }_{X}})-\frac{1}{2}\sigma _{Y}^{2}\).