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Student[Statistics]

 NormalRandomVariable
 normal (Gaussian) random variable

 Calling Sequence NormalRandomVariable(mu, sigma)

Parameters

 mu - distribution mean sigma - scale parameter

Description

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

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

 subject to the following conditions:

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

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

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{Statistics}]\right):$
 > $X≔\mathrm{NormalRandomVariable}\left(\mathrm{μ},\mathrm{σ}\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}{{ⅇ}}^{{-}\frac{{1}}{{2}}{}\frac{{\left({u}{-}{\mathrm{μ}}\right)}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}}{\sqrt{{\mathrm{π}}}{}{\mathrm{σ}}}$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{0.3989422802}{}{{ⅇ}}^{{-}\frac{{0.5000000000}{}{\left({0.5}{-}{1.}{}{\mathrm{μ}}\right)}^{{2}}}{{{\mathrm{σ}}}^{{2}}}}}{{\mathrm{σ}}}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 ${\mathrm{μ}}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 ${{\mathrm{σ}}}^{{2}}$ (4)
 > $Y≔\mathrm{NormalRandomVariable}\left(3,5\right):$
 > $\mathrm{PDF}\left(Y,x,\mathrm{output}=\mathrm{plot}\right)$
 > $\mathrm{CDF}\left(Y,x\right)$
 $\frac{{1}}{{2}}{+}\frac{{1}}{{2}}{}{\mathrm{erf}}{}\left(\frac{{1}}{{10}}{}\left({x}{-}{3}\right){}\sqrt{{2}}\right)$ (5)
 > $\mathrm{CDF}\left(Y,3,\mathrm{output}=\mathrm{plot}\right)$

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][NormalRandomVariable] command was introduced in Maple 18.