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The normal distribution is a continuous probability distribution 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.

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

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

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

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

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

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

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

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

$\mathrm{\mu }$

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

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

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.