InverseSurvivalFunction - Maple Help

Statistics

 InverseSurvivalFunction
 compute the inverse survival function

 Calling Sequence InverseSurvivalFunction(X, t, options)

Parameters

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

Description

 • The InverseSurvivalFunction function computes the inverse survival 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 inverse survival function is computed using exact arithmetic. To compute the inverse survival function numerically, specify the numeric or numeric = true option.

Examples

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

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

 > $\mathrm{InverseSurvivalFunction}\left(\mathrm{Cauchy}\left(p,q\right),t\right)$
 ${p}{+}{q}{}{\mathrm{tan}}{}\left({\mathrm{\pi }}{}\left(\frac{{1}}{{2}}{-}{t}\right)\right)$ (1)

Use numeric parameters.

 > $\mathrm{InverseSurvivalFunction}\left(\mathrm{Cauchy}\left(3,5\right),\frac{1}{2}\right)$
 ${3}$ (2)
 > $\mathrm{InverseSurvivalFunction}\left(\mathrm{Cauchy}\left(3,5\right),\frac{1}{2},\mathrm{numeric}\right)$
 ${3.}$ (3)
 > $\mathrm{Quantile}\left(\mathrm{Cauchy}\left(3,5\right),\frac{1}{2}\right)$
 ${3}$ (4)

Define new distribution.

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

Another distribution

 > $U≔\mathrm{Distribution}\left(\mathrm{=}\left(\mathrm{CDF},t↦F\left(t\right)\right),\mathrm{=}\left(\mathrm{PDF},t↦f\left(t\right)\right)\right):$
 > $Y≔\mathrm{RandomVariable}\left(U\right):$
 > $\mathrm{CDF}\left(Y,t\right)$
 ${F}{}\left({t}\right)$ (7)
 > $\mathrm{InverseSurvivalFunction}\left(Y,t\right)$
 ${\mathrm{RootOf}}{}\left({F}{}\left({\mathrm{_Z}}\right){-}{1}{+}{t}\right)$ (8)

References

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