ProbabilityDensityFunction - Maple Help

Statistics

 ProbabilityDensityFunction
 compute the probability density function

 Calling Sequence ProbabilityDensityFunction(X, t, options) PDF(X, t, options)

Parameters

 X - algebraic; random variable or distribution t - algebraic; point options - (optional) equations; specify options for computing the probability density function of a random variable

Description

 • 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]).

Computation

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

Options

 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.

Examples

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

Compute the probability density function of the beta distribution with parameters $p$ and $q$.

 > $\mathrm{ProbabilityDensityFunction}\left('\mathrm{Β}'\left(p,q\right),t\right)$
 $\left\{\begin{array}{cc}{0}& {t}{<}{0}\\ \frac{{{t}}^{{-}{1}{+}{p}}{}{\left({1}{-}{t}\right)}^{{-}{1}{+}{q}}}{{\mathrm{Β}}{}\left({p}{,}{q}\right)}& {t}{<}{1}\\ {0}& {\mathrm{otherwise}}\end{array}\right\$ (1)

Use numeric parameters.

 > $\mathrm{ProbabilityDensityFunction}\left('\mathrm{Β}'\left(3,5\right),\frac{1}{2}\right)$
 $\frac{{105}}{{64}}$ (2)
 > $\mathrm{ProbabilityDensityFunction}\left('\mathrm{Β}'\left(3,5\right),\frac{1}{2},\mathrm{numeric}\right)$
 ${1.640625000}$ (3)

Define new distribution.

 > $T≔\mathrm{Distribution}\left(\mathrm{=}\left(\mathrm{PDF},t↦\frac{1}{\mathrm{\pi }\cdot \left({t}^{2}+1\right)}\right)\right):$
 > $X≔\mathrm{RandomVariable}\left(T\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\frac{{1}}{{\mathrm{\pi }}{}\left({{u}}^{{2}}{+}{1}\right)}$ (4)
 > $\mathrm{PDF}\left(X,0\right)$
 $\frac{{1}}{{\mathrm{\pi }}}$ (5)
 > $\mathrm{CDF}\left(X,u\right)$
 $\frac{{\mathrm{\pi }}{+}{2}{}{\mathrm{arctan}}{}\left({u}\right)}{{2}{}{\mathrm{\pi }}}$ (6)

Use the inert option with a new RandomVariable, $Y$.

 > $Y≔\mathrm{RandomVariable}\left(\mathrm{Distribution}\left(\mathrm{=}\left(\mathrm{CDF},u↦\frac{\mathrm{\pi }+2\cdot \mathrm{arctan}\left(u\right)}{2\cdot \mathrm{\pi }}\right)\right)\right)$
 ${Y}{≔}{\mathrm{_R3}}$ (7)
 > $\mathrm{PDF}\left(Y,t\right)$
 $\frac{{1}}{\left({{t}}^{{2}}{+}{1}\right){}{\mathrm{\pi }}}$ (8)
 > $\mathrm{PDF}\left(Y,t,\mathrm{inert}\right)$
 $\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left(\frac{{\mathrm{\pi }}{+}{2}{}{\mathrm{arctan}}{}\left({t}\right)}{{2}{}{\mathrm{\pi }}}\right)$ (9)

References

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