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

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 $q$.

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

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

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

 > $Y≔\mathrm{RandomVariable}\left(\mathrm{Distribution}\left(\mathrm{CDF}=\left(u→\frac{1\left(\mathrm{π}+2\mathrm{arctan}\left(u\right)\right)}{2\mathrm{π}}\right)\right)\right)$
 ${Y}{≔}{\mathrm{_R3}}$ (7)
 > $\mathrm{PDF}\left(Y,t\right)$
 $\frac{{1}}{\left({{t}}^{{2}}{+}{1}\right){}{\mathrm{π}}}$ (8)
 > $\mathrm{PDF}\left(Y,t,\mathrm{inert}\right)$
 $\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{1}}{{2}}{}\frac{{\mathrm{π}}{+}{2}{}{\mathrm{arctan}}{}\left({t}\right)}{{\mathrm{π}}}\right)$ (9)

References

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