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The log normal distribution is a continuous probability distribution with probability density function given by:

subject to the following conditions:

The LogNormal variate with mean log parameter mu and scale parameter sigma is related to the Normal variate by LogNormal(mu,sigma) ~ exp(Normal(mu,sigma)).

Note that the LogNormal command is inert and should be used in combination with the RandomVariable command.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{LogNormal}\left(\mathrm{\mu }\,\mathrm{\sigma }\right)\right)\:$

$\mathrm{PDF}\left(X\,u\right)$

$\left\{\begin{array}{cc}0& u<0\\ \frac{\sqrt{2}{\&ExponentialE;}^{-\frac{{\left(\ln \left(u\right)-\mathrm{\mu }\right)}^2}{2{\mathrm{\sigma }}^2}}}{2u\mathrm{\sigma }\sqrt{\mathrm{\pi }}}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{PDF}\left(X\&comma;0.5\right)$

$\frac{0.7978845605{\&ExponentialE;}^{-\frac{0.5000000000{\left(-0.6931471806-1.\mathrm{\mu }\right)}^2}{{\mathrm{\sigma }}^2}}}{\mathrm{\sigma }}$

$\mathrm{Mean}\left(X\right)$

${\&ExponentialE;}^{\mathrm{\mu }+\frac{{\mathrm{\sigma }}^2}{2}}$

$\mathrm{Variance}\left(X\right)$

${\&ExponentialE;}^{{\mathrm{\sigma }}^2+2\mathrm{\mu }}\left({\&ExponentialE;}^{{\mathrm{\sigma }}^2}-1\right)$

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.

Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.