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Student[Statistics]

 LogNormalRandomVariable
 log normal random variable

 Calling Sequence LogNormalRandomVariable(mu, sigma)

Parameters

 mu - mean log parameter sigma - scale parameter

Description

 • The log normal random variable is a continuous probability random variable 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)).

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{Statistics}]\right):$
 > $X≔\mathrm{LogNormalRandomVariable}\left(\mathrm{μ},\mathrm{σ}\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 ${{}\begin{array}{cc}{0}& {u}{<}{0}\\ \frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}{{ⅇ}}^{{-}\frac{{1}}{{2}}{}\frac{{\left({\mathrm{ln}}{}\left({u}\right){-}{\mathrm{μ}}\right)}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}}{{u}{}{\mathrm{σ}}{}\sqrt{{\mathrm{π}}}}& {\mathrm{otherwise}}\end{array}$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{0.7978845605}{}{{ⅇ}}^{{-}\frac{{0.5000000000}{}{\left({-}{0.6931471806}{-}{1.}{}{\mathrm{μ}}\right)}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}}{{\mathrm{σ}}}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 ${{ⅇ}}^{{\mathrm{μ}}{+}\frac{{1}}{{2}}{}{{\mathrm{σ}}}^{{2}}}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 ${{ⅇ}}^{{{\mathrm{σ}}}^{{2}}{+}{2}{}{\mathrm{μ}}}{}\left({{ⅇ}}^{{{\mathrm{σ}}}^{{2}}}{-}{1}\right)$ (4)
 > $Y≔\mathrm{LogNormalRandomVariable}\left(1,2\right):$
 > $\mathrm{PDF}\left(Y,x,\mathrm{output}=\mathrm{plot}\right)$
 > $\mathrm{CDF}\left(Y,x\right)$
 ${{}\begin{array}{cc}{0}& {x}{<}{0}\\ \frac{{1}}{{2}}{+}\frac{{1}}{{2}}{}{\mathrm{erf}}{}\left(\frac{{1}}{{4}}{}\left({\mathrm{ln}}{}\left({x}\right){-}{1}\right){}\sqrt{{2}}\right)& {\mathrm{otherwise}}\end{array}$ (5)
 > $\mathrm{CDF}\left(Y,4,\mathrm{output}=\mathrm{plot}\right)$

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.

Compatibility

 • The Student[Statistics][LogNormalRandomVariable] command was introduced in Maple 18.