The ProbabilityDensityFunction function computes the probability density function of the specified random variable at the specified point.

The first parameter can be a distribution (see Statistics[Distribution]), a random variable, or an algebraic expression involving random variables (see Statistics[RandomVariable]).

By default, all computations involving random variables are performed symbolically (see option numeric below).

For more information about computation in the Statistics package, see the Statistics[Computation] help page.

The options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[RandomVariables] help page.

numeric=truefalse -- By default, the probability density function is computed using exact arithmetic. To compute the probability density function numerically, specify the numeric or numeric = true option.

inert=truefalse -- By default, Maple evaluates integrals, sums, derivatives and limits encountered while computing the PDF. By specifying inert or inert=true, Maple will return these unevaluated.

mainbranch - returns the main branch of the distribution only.

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

$\mathrm{ProbabilityDensityFunction}\left('\mathrm{{\rm B}}'\left(p\,q\right)\,t\right)$

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

$\mathrm{ProbabilityDensityFunction}\left('\mathrm{{\rm B}}'\left(3\&comma;5\right)\&comma;\frac{1}{2}\right)$

$\frac{105}{64}$

$\mathrm{ProbabilityDensityFunction}\left('\mathrm{{\rm B}}'\left(3\&comma;5\right)\&comma;\frac{1}{2}\&comma;\mathrm{numeric}\right)$

$1.640625000$

$T≔\mathrm{Distribution}\left(\mathrm{`=`}\left(\mathrm{PDF}\&comma;t\mapsto \frac{1}{\mathrm{\pi }\cdot \left(t^2+1\right)}\right)\right)\&colon;$

$X≔\mathrm{RandomVariable}\left(T\right)\&colon;$

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

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

$\mathrm{PDF}\left(X\&comma;0\right)$

$\frac{1}{\mathrm{\pi }}$

$\mathrm{CDF}\left(X\&comma;u\right)$

$\frac{\mathrm{\pi }+2\arctan \left(u\right)}{2\mathrm{\pi }}$

$Y≔\mathrm{RandomVariable}\left(\mathrm{Distribution}\left(\mathrm{`=`}\left(\mathrm{CDF}\&comma;u\mapsto \frac{\mathrm{\pi }+2\cdot \arctan \left(u\right)}{2\cdot \mathrm{\pi }}\right)\right)\right)$

$Y≔\mathrm{\_R3}$

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

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

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

$\frac{\&DifferentialD;}{\&DifferentialD;t}\left(\frac{\mathrm{\pi }+2\arctan \left(t\right)}{2\mathrm{\pi }}\right)$

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.