CumulativeDistributionFunction - Maple Help

Statistics

 CumulativeDistributionFunction
 compute the cumulative distribution function

 Calling Sequence CumulativeDistributionFunction(X, t, options) CDF(X, t, options)

Parameters

 X - algebraic; random variable or distribution t - algebraic; point options - (optional) equation(s) of the form numeric=value or inert=value; specifies options for computing the cumulative distribution function of a random variable

Description

 • The CumulativeDistributionFunction function computes the cumulative distribution 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]).
 • The inverse function of the CDF is the Quantile.

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 cumulative distribution function is computed using exact arithmetic. To compute the cumulative distribution function numerically, specify the numeric or numeric=true option.
 • inert=truefalse -- By default, Maple evaluates integrals, sums, derivatives and limits encountered while computing the CDF. By specifying inert or inert=true, Maple will return these unevaluated.

Examples

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

Compute the cumulative distribution function of the beta distribution with parameters p and q.

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

Use numeric parameters.

 > $\mathrm{CumulativeDistributionFunction}\left('\mathrm{Β}'\left(3,5\right),\frac{1}{2}\right)$
 $\frac{{35}{}{\mathrm{hypergeom}}{}\left(\left[{-4}{,}{3}\right]{,}\left[{4}\right]{,}\frac{{1}}{{2}}\right)}{{8}}$ (2)
 > $\mathrm{CumulativeDistributionFunction}\left('\mathrm{Β}'\left(3,5\right),\frac{1}{2},\mathrm{numeric}\right)$
 ${0.773437500000000}$ (3)

Define new distribution.

 > $T≔\mathrm{Distribution}\left(\mathrm{=}\left(\mathrm{PDF},t↦\frac{1}{\mathrm{\pi }\cdot \left({t}^{2}+1\right)}\right)\right):$
 > $X≔\mathrm{RandomVariable}\left(T\right):$
 > $\mathrm{CDF}\left(X,0\right)$
 $\frac{{1}}{{2}}$ (4)
 > $\mathrm{CDF}\left(X,0,\mathrm{numeric}\right)$
 ${0.4999999999}$ (5)
 > $\mathrm{CDF}\left(X,u\right)$
 $\frac{{\mathrm{\pi }}{+}{2}{}{\mathrm{arctan}}{}\left({u}\right)}{{2}{}{\mathrm{\pi }}}$ (6)

Use the inert option.

 > $\mathrm{CDF}\left(X,0,\mathrm{inert}=\mathrm{true}\right)$
 ${{\int }}_{{-}{\mathrm{\infty }}}^{{0}}\frac{{1}}{{\mathrm{\pi }}{}\left({{\mathrm{_t}}}^{{2}}{+}{1}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_t}}$ (7)
 > $\mathrm{CDF}\left(X,t,\mathrm{inert}=\mathrm{true}\right)$
 ${{\int }}_{{-}{\mathrm{\infty }}}^{{t}}\frac{{1}}{{\mathrm{\pi }}{}\left({{\mathrm{_t0}}}^{{2}}{+}{1}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_t0}}$ (8)
 > $N≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(0,1\right)\right):$
 > $\mathrm{CDF}\left(N,t,\mathrm{inert}=\mathrm{true}\right)$
 ${{\int }}_{{-}{\mathrm{\infty }}}^{{t}}\frac{\sqrt{{2}}{}{{ⅇ}}^{{-}\frac{{{\mathrm{_t1}}}^{{2}}}{{2}}}}{{2}{}\sqrt{{\mathrm{\pi }}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_t1}}$ (9)

References

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