Random variables are created using the RandomVariable command.

with(Statistics):

X := RandomVariable(Normal(5, 1));

$X≔\mathrm{\_R}$

Y := RandomVariable(Normal(7, 1));

$Y≔\mathrm{\_R0}$

Random variables can be distinguished from ordinary variables (names) by their attributes. Type RandomVariable can be used to query whether a given Maple object is a random variable or not.

type(X, RandomVariable);

$\mathrm{true}$

type(Z, RandomVariable);

$\mathrm{false}$

indets(X+Y+Z, RandomVariable);

$\left\{\mathrm{\_R}\,\mathrm{\_R0}\right\}$

The Statistics package provides a number of tools for computing basic quantities and functions. Single random variables as well as algebraic expressions (e.g. linear combinations, products, etc.) involving random variables are supported. Different random variables involved in an expression are considered to be independent. By default, all computations involving random variables are performed symbolically.

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

$\mathrm{PDF}\left(\mathrm{NonCentralBeta}\left(5\,3\,m\right)\,t\right)$

$\left\{\begin{array}{cc}0& t<0\\ {\&ExponentialE;}^{-\frac{m}{2}}{t}^4{\left(1-t\right)}^2\left(\frac{1}{16}{m}^3{t}^3+\frac{21}{8}{m}^2{t}^2+\frac{63}{2}mt+105\right){\&ExponentialE;}^{\frac{mt}{2}}& t\le 1\\ 0& \mathrm{otherwise}\end{array}\right.$

$\mathrm{CDF}\left(\mathrm{NonCentralBeta}\left(5\&comma;3\&comma;m\right)\&comma;t\right)$

$\left\{\begin{array}{cc}0& t\le 0\\ \frac{t^5{\&ExponentialE;}^{-\frac{1}{2}m+\frac{1}{2}mt}\left(m^2{t}^4-2m^2{t}^3+m^2{t}^2+24m{t}^3-52m{t}^2+28mt+120t^2-280t+168\right)}{8}& t\le 1\\ 1& 1<t\end{array}\right.$

$X≔\mathrm{RandomVariable}\left(\mathrm{{\rm B}}\left(p\&comma;q\right)\right)$

$X≔\mathrm{\_R3}$

$\mathrm{PDF}\left(X\&comma;t\right)$

$\left\{\begin{array}{cc}0& t<0\\ \frac{t^{p-1}{\left(1-t\right)}^{q-1}}{\mathrm{{\rm B}}\left(p\&comma;q\right)}& t<1\\ 0& \mathrm{otherwise}\end{array}\right.$

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

$\frac{p}{p+q}$

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

$\frac{\sqrt{\frac{pq}{p+q+1}}}{p+q}$

$\mathrm{Moment}\left(X\&comma;n\right)$

$\frac{\mathrm{{\rm B}}\left(q\&comma;n+p\right)}{\mathrm{{\rm B}}\left(p\&comma;q\right)}$

$X≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(0\&comma;1\right)\right)$

$X≔\mathrm{\_R4}$

$Y≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(0\&comma;1\right)\right)$

$Y≔\mathrm{\_R5}$

$\mathrm{PDF}\left(\frac{X}{Y}\&comma;t\right)$

$\frac{1}{\left(t^2+1\right)\mathrm{\pi }}$

$\mathrm{PDF}\left(\mathrm{Cauchy}\left(0\&comma;1\right)\&comma;t\right)$

$\frac{1}{\left(t^2+1\right)\mathrm{\pi }}$

$X≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(0\&comma;1\right)\right)\&colon;$

$\mathrm{Probability}\left(X^2<X\right)$

$\frac{\mathrm{erf}\left(\frac{\sqrt{2}}{2}\right)}{2}$

$\mathrm{assume}\left(0<m\&comma;0<k\&comma;0<T\right)$

$\mathrm{\sigma }≔\mathrm{sqrt}\left(\frac{kT}{m}\right)$

$\mathrm{\sigma }≔\sqrt{\frac{\mathrm{k~}\mathrm{T~}}{\mathrm{m~}}}$

$f≔\mathrm{simplify}\left(\mathrm{piecewise}\left(x<0\&comma;0\&comma;\frac{\mathrm{sqrt}\left(\frac{2}{\mathrm{\pi }}\right)}{{\mathrm{\sigma }}^3}{x}^2\exp \left(-\frac{x^2}{2{\mathrm{\sigma }}^2}\right)\right)\right)$

$f≔\left\{\begin{array}{cc}0& x<0\\ \frac{\sqrt{2}{\mathrm{m~}}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{x}^2{\&ExponentialE;}^{-\frac{x^2\mathrm{m~}}{2\mathrm{k~}\mathrm{T~}}}}{\sqrt{\mathrm{\pi }}{\mathrm{k~}}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\mathrm{T~}}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}& 0\le x\end{array}\right.$

$\mathrm{MD}≔\mathrm{Distribution}\left(\mathrm{PDF}=\mathrm{unapply}\left(f\&comma;x\right)\right)$

$\mathrm{MD}≔\mathbf{module}\left(\right)\mathbf{option}\mathrm{Distribution}\&comma;\mathrm{Continuous}\&semi;\mathbf{export}\mathrm{PDF}\&comma;\mathrm{Conditions}\&semi;\mathbf{end\; module}$

$X≔\mathrm{RandomVariable}\left(\mathrm{MD}\right)$

$X≔\mathrm{\_R8}$

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

$\frac{2\sqrt{\mathrm{T~}}\sqrt{\mathrm{k~}}\sqrt{2}}{\sqrt{\mathrm{m~}}\sqrt{\mathrm{\pi }}}$

$\mathrm{simplify}\left(\mathrm{ExpectedValue}\left(\frac{1}{2}m{X}^2\right)\right)$

$\frac{3\mathrm{k~}\mathrm{T~}}{2}$

$g≔\mathrm{eval}\left(f\&comma;\left[T=298.15\&comma;k=\mathrm{evalf}\left(\mathrm{ScientificConstants}:-\mathrm{Constant}\left('k'\right)\right)\&comma;m=4.\times {10}^{\mathrm{-27}}\right]\right)$

$g≔\left\{\begin{array}{cc}0& x<0\\ 5.404283672\times {10}^{\mathrm{-10}}\sqrt{2}{x}^2{\&ExponentialE;}^{-4.858610154\times {10}^{\mathrm{-7}}{x}^2}& 0\le x\end{array}\right.$

$\mathrm{He}≔\mathrm{Distribution}\left(\mathrm{PDF}=\mathrm{unapply}\left(g\&comma;x\right)\right)\&colon;$

$\mathrm{XHe}≔\mathrm{RandomVariable}\left(\mathrm{He}\right)\&colon;$

$\mathrm{Mode}\left(\mathrm{XHe}\right)$

$\left\{1434.643427\right\}$

$A≔\mathrm{Sample}\left(\mathrm{XHe}\&comma;{10}^5\right)$

$\mathrm{Mode}\left(A\right)$

$1444.17684604007$

$P≔\mathrm{DensityPlot}\left(\mathrm{XHe}\&comma;\mathrm{range}=0..3500\&comma;\mathrm{thickness}=3\&comma;\mathrm{color}=\mathrm{red}\right)\&colon;$

$Q≔\mathrm{Histogram}\left(A\&comma;\mathrm{range}=0..3500\right)\&colon;$

$\mathrm{plots}\left[\mathrm{display}\right]\left(P\&comma;Q\right)$