The CumulativeDistributionFunction function computes the cumulative distribution function of the specified random variable at the specified point.

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

The inverse function of the CDF is the Quantile.

If the option output is not included or is specified to be output=value, then the function will return the value of the cumulative density function of the specified random variable at the specified point. If output=plot is specified, then the function will return a plot of CDF(X,x) together with dash lines instructing the value at x = specified point. If output=both is specified, then both the value and the plot will be returned.

If the option inert is not included or is specified to be inert=false, then the function will return the actual value of the result. If inert or inert=true is specified, then the function will return the formula of evaluating the actual value.

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

If the second parameter is a floating point value or the option numeric is included, then the computation is done in floating point. Otherwise the computation is exact.

By default, the CDF of the specified random variable at the specified point is computed according to the rules mentioned above. To always compute the value numerically, specify the numeric or numeric = true option.

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

$\mathrm{CumulativeDistributionFunction}\left(\mathrm{BetaRandomVariable}\left(p\,q\right)\,t\right)$

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

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

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

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

$0.773437500000000$

$X≔{\mathrm{NormalRandomVariable}\left(2\&comma;5\right)}^2$

$X≔{\mathrm{\_R2}}^2$

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

$\left\{\begin{array}{cc}0& x\le 0\\ \frac{\mathrm{erf}\left(\frac{\sqrt{2}\sqrt{x}}{10}+\frac{\sqrt{2}}{5}\right)}{2}+\frac{\mathrm{erf}\left(\frac{\sqrt{2}\sqrt{x}}{10}-\frac{\sqrt{2}}{5}\right)}{2}& 0<x\end{array}\right.$

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

${\int }_{-\mathrm{\infty }}^x\left(\left\{\begin{array}{cc}0& \mathrm{\_t}\le 0\\ \frac{\sqrt{2}\left({\&ExponentialE;}^{-\frac{{\left(\sqrt{\mathrm{\_t}}+2\right)}^2}{50}}+{\&ExponentialE;}^{-\frac{{\left(\sqrt{\mathrm{\_t}}-2\right)}^2}{50}}\right)}{20\sqrt{\mathrm{\_t}}\sqrt{\mathrm{\pi }}}& 0<\mathrm{\_t}\end{array}\right.\right)\&DifferentialD;\mathrm{\_t}$

$\mathrm{CDF}\left(X\&comma;x\&comma;\mathrm{output}=\mathrm{plot}\right)$

$P≔\mathrm{PoissonRandomVariable}\left(3\right)\&colon;$

$\mathrm{cdf},\mathrm{graph}≔\mathrm{CDF}\left(P\&comma;x\&comma;\mathrm{output}=\mathrm{both}\right)\&colon;$

$\mathrm{cdf}$

$1-\frac{33^{\max \left(\mathrm{-1}\&comma;⌊x⌋\right)}}{3^{\max \left(\mathrm{-1}\&comma;⌊x⌋\right)+1}}+\frac{3\mathrm{\Gamma }\left(\max \left(\mathrm{-1}\&comma;⌊x⌋\right)+1\&comma;3\right)3^{\max \left(\mathrm{-1}\&comma;⌊x⌋\right)}}{3^{\max \left(\mathrm{-1}\&comma;⌊x⌋\right)+1}\mathrm{\Gamma }\left(\max \left(\mathrm{-1}\&comma;⌊x⌋\right)+1\right)}$

$\mathrm{graph}$

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

The Student[Statistics][CumulativeDistributionFunction] and Student[Statistics][CDF] commands were introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.