 SurvivalFunction - Maple Help

Statistics

 SurvivalFunction
 compute the survival function Calling Sequence SurvivalFunction(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 survival function of a random variable Description

 • The SurvivalFunction function computes the survival function of the random variable X at the point t, which is defined as the probability that X takes a value greater than t. In other words, if $S\left(t\right)$ denotes the survival function of X and $F\left(t\right)$ denotes the cumulative distribution function of X, then $S\left(t\right)=1-F\left(t\right)$ for all real values of t.
 • 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). 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 survival function is computed using exact arithmetic. To compute the survival function numerically, specify the numeric or numeric = true option. Examples

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

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

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

If p = 3 and q = 5, the plot of the survival function is as follows:

 > $\mathrm{plot}\left(\mathrm{SurvivalFunction}\left('\mathrm{Β}'\left(3,5\right),t\right),t=0..1\right)$ The survival function can also be evaluated directly using numeric parameters.

 > $\mathrm{SurvivalFunction}\left('\mathrm{Β}'\left(3,5\right),\frac{1}{2}\right)$
 ${1}{-}\frac{{35}{}{\mathrm{hypergeom}}{}\left(\left[{-4}{,}{3}\right]{,}\left[{4}\right]{,}\frac{{1}}{{2}}\right)}{{8}}$ (2)
 > $\mathrm{simplify}\left(\right)$
 $\frac{{29}}{{128}}$ (3)

The numeric option gives a floating point result.

 > $\mathrm{SurvivalFunction}\left('\mathrm{Β}'\left(3,5\right),\frac{1}{2},\mathrm{numeric}\right)$
 ${0.226562500000000}$ (4)

Define new distribution.

 > $T≔\mathrm{Distribution}\left(\mathrm{PDF}=\left(t→\frac{1}{\mathrm{Pi}\left({t}^{2}+1\right)}\right)\right):$
 > $X≔\mathrm{RandomVariable}\left(T\right):$
 > $\mathrm{CDF}\left(X,t\right)$
 $\frac{{\mathrm{\pi }}{+}{2}{}{\mathrm{arctan}}{}\left({t}\right)}{{2}{}{\mathrm{\pi }}}$ (5)
 > $\mathrm{SurvivalFunction}\left(X,t\right)$
 ${1}{-}\frac{{\mathrm{\pi }}{+}{2}{}{\mathrm{arctan}}{}\left({t}\right)}{{2}{}{\mathrm{\pi }}}$ (6)
 > $\mathrm{plot}\left(,t=-10..10\right)$ Another distribution

 > $U≔\mathrm{Distribution}\left(\mathrm{CDF}=\left(t→F\left(t\right)\right),\mathrm{PDF}=\left(t→f\left(t\right)\right)\right):$
 > $Y≔\mathrm{RandomVariable}\left(U\right):$
 > $\mathrm{CDF}\left(Y,t\right)$
 ${F}{}\left({t}\right)$ (7)
 > $\mathrm{SurvivalFunction}\left(Y,t\right)$
 ${1}{-}{F}{}\left({t}\right)$ (8) References

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