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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 .

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 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.

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

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

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

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

$\frac{35\mathrm{hypergeom}\left(\left[\mathrm{-4}\&comma;3\right]\&comma;\left[4\right]\&comma;\frac{1}{2}\right)}{8}$

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

$0.773437500000000$

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

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

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

$\frac{1}{2}$

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

$0.4999999999$

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

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

$\mathrm{CDF}\left(X\&comma;0\&comma;\mathrm{inert}=\mathrm{true}\right)$

${\int }_{-\mathrm{\infty }}^0\frac{1}{\mathrm{\pi }\left({\mathrm{\_t}}^2+1\right)}\&DifferentialD;\mathrm{\_t}$

$\mathrm{CDF}\left(X\&comma;t\&comma;\mathrm{inert}=\mathrm{true}\right)$

${\int }_{-\mathrm{\infty }}^t\frac{1}{\mathrm{\pi }\left({\mathrm{\_t0}}^2+1\right)}\&DifferentialD;\mathrm{\_t0}$

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

$\mathrm{CDF}\left(N\&comma;t\&comma;\mathrm{inert}=\mathrm{true}\right)$

${\int }_{-\mathrm{\infty }}^t\frac{\sqrt{2}{\&ExponentialE;}^{-\frac{{\mathrm{\_t1}}^2}{2}}}{2\sqrt{\mathrm{\pi }}}\&DifferentialD;\mathrm{\_t1}$

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