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]).

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

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

$\mathrm{InverseSurvivalFunction}\left(\mathrm{Cauchy}\left(p\,q\right)\,t\right)$

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

$\mathrm{InverseSurvivalFunction}\left(\mathrm{Cauchy}\left(3\,5\right)\,\frac{1}{2}\right)$

$3$

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

$3.$

$\mathrm{Quantile}\left(\mathrm{Cauchy}\left(3\,5\right)\,\frac{1}{2}\right)$

$3$

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

$X≔\mathrm{RandomVariable}\left(T\right)\:$

$\mathrm{CDF}\left(X\,t\right)$

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

$\mathrm{InverseSurvivalFunction}\left(X\,t\right)$

$-\cot \left(\left(1-t\right)\mathrm{\pi }\right)$

$U≔\mathrm{Distribution}\left(\mathrm{`=`}\left(\mathrm{CDF}\,t\mapsto F\left(t\right)\right)\,\mathrm{`=`}\left(\mathrm{PDF}\,t\mapsto f\left(t\right)\right)\right)\:$

$Y≔\mathrm{RandomVariable}\left(U\right)\:$

$\mathrm{CDF}\left(Y\,t\right)$

$F\left(t\right)$

$\mathrm{InverseSurvivalFunction}\left(Y\,t\right)$

$\mathrm{RootOf}\left(F\left(\mathrm{\_Z}\right)-1+t\right)$

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