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Statistics[Distributions]

 LogNormal
 log normal distribution

 Calling Sequence LogNormal(mu, sigma) LogNormalDistribution(mu, sigma)

Parameters

 mu - mean log parameter sigma - scale parameter

Description

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

$f\left(t\right)=\left\{\begin{array}{cc}0& t<0\\ \frac{\sqrt{2}{ⅇ}^{-\frac{{\left(\mathrm{ln}\left(t\right)-\mathrm{\mu }\right)}^{2}}{2{\mathrm{\sigma }}^{2}}}}{2t\mathrm{\sigma }\sqrt{\mathrm{\pi }}}& \mathrm{otherwise}\end{array}\right\$

 subject to the following conditions:

$\mathrm{\mu }::\mathrm{real},0<\mathrm{\sigma }$

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{LogNormal}\left(\mathrm{\mu },\mathrm{\sigma }\right)\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\left\{\begin{array}{cc}{0}& {u}{<}{0}\\ \frac{\sqrt{{2}}{}{{ⅇ}}^{{-}\frac{{\left({\mathrm{ln}}{}\left({u}\right){-}{\mathrm{\mu }}\right)}^{{2}}}{{2}{}{{\mathrm{\sigma }}}^{{2}}}}}{{2}{}{u}{}{\mathrm{\sigma }}{}\sqrt{{\mathrm{\pi }}}}& {\mathrm{otherwise}}\end{array}\right\$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{0.7978845605}{}{{ⅇ}}^{{-}\frac{{0.5000000000}{}{\left({-}{0.6931471806}{-}{1.}{}{\mathrm{\mu }}\right)}^{{2}}}{{{\mathrm{\sigma }}}^{{2}}}}}{{\mathrm{\sigma }}}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 ${{ⅇ}}^{{\mathrm{\mu }}{+}\frac{{{\mathrm{\sigma }}}^{{2}}}{{2}}}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 ${{ⅇ}}^{{{\mathrm{\sigma }}}^{{2}}{+}{2}{}{\mathrm{\mu }}}{}\left({{ⅇ}}^{{{\mathrm{\sigma }}}^{{2}}}{-}{1}\right)$ (4)

References

 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.