The normal random variable is a continuous probability random variable with probability density function given by:

subject to the following conditions:

The normal variate Normal(mu,sigma) is related to the standardized variate Normal(0,1) by Normal(0,1) ~ (Normal(mu,sigma)-mu)/sigma.

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

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

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

$\frac{\sqrt{2}{\ⅇ}^{-\frac{{\left(u-\mathrm{\mu }\right)}^2}{2{\mathrm{\sigma }}^2}}}{2\sqrt{\mathrm{\pi }}\mathrm{\sigma }}$

$\mathrm{PDF}\left(X\,0.5\right)$

$\frac{0.3989422802{\ⅇ}^{-\frac{0.5000000000{\left(0.5-1.\mathrm{\mu }\right)}^2}{{\mathrm{\sigma }}^2}}}{\mathrm{\sigma }}$

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

$\mathrm{\mu }$

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

${\mathrm{\sigma }}^2$

$Y≔\mathrm{NormalRandomVariable}\left(3\,5\right)\:$

$\mathrm{PDF}\left(Y\,x\,\mathrm{output}=\mathrm{plot}\right)$

$\mathrm{CDF}\left(Y\,x\right)$

$\frac{1}{2}+\frac{\mathrm{erf}\left(\frac{\left(x-3\right)\sqrt{2}}{10}\right)}{2}$

$\mathrm{CDF}\left(Y\,3\,\mathrm{output}=\mathrm{plot}\right)$

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.

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

For more information on Maple 18 changes, see Updates in Maple 18.