MillsRatio - Maple Help
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Statistics

 MillsRatio
 compute the Mills ratio

 Calling Sequence MillsRatio(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 Mills ratio of a random variable

Description

 • The MillsRatio ratio computes the Mills ratio 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 Mills ratio is computed using exact arithmetic. To compute the Mills ratio numerically, specify the numeric or numeric = true option.

Examples

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

Compute the Mills ratio of the beta distribution with parameters p and q.

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

Use numeric parameters.

 > $\mathrm{MillsRatio}\left('\mathrm{Β}'\left(3,5\right),\frac{1}{2}\right)$
 $\frac{{64}}{{105}}{-}\frac{{8}{}{\mathrm{hypergeom}}{}\left(\left[{-4}{,}{3}\right]{,}\left[{4}\right]{,}\frac{{1}}{{2}}\right)}{{3}}$ (2)
 > $\mathrm{MillsRatio}\left('\mathrm{Β}'\left(3,5\right),\frac{1}{2},\mathrm{numeric}\right)$
 ${0.138095238095238}$ (3)

Define new distribution.

 > $T≔\mathrm{Distribution}\left(\mathrm{=}\left(\mathrm{PDF},t↦\frac{1}{\mathrm{\pi }\cdot \left({t}^{2}+1\right)}\right)\right):$
 > $X≔\mathrm{RandomVariable}\left(T\right):$
 > $\mathrm{CDF}\left(X,t\right)$
 $\frac{{\mathrm{\pi }}{+}{2}{}{\mathrm{arctan}}{}\left({t}\right)}{{2}{}{\mathrm{\pi }}}$ (4)
 > $\mathrm{MillsRatio}\left(X,t\right)$
 $\left({1}{-}\frac{{\mathrm{\pi }}{+}{2}{}{\mathrm{arctan}}{}\left({t}\right)}{{2}{}{\mathrm{\pi }}}\right){}{\mathrm{\pi }}{}\left({{t}}^{{2}}{+}{1}\right)$ (5)

Another distribution

 > $U≔\mathrm{Distribution}\left(\mathrm{=}\left(\mathrm{CDF},t↦F\left(t\right)\right),\mathrm{=}\left(\mathrm{PDF},t↦f\left(t\right)\right)\right):$
 > $Y≔\mathrm{RandomVariable}\left(U\right):$
 > $\mathrm{CDF}\left(Y,t\right)$
 ${F}{}\left({t}\right)$ (6)
 > $\mathrm{MillsRatio}\left(Y,t\right)$
 $\frac{{1}{-}{F}{}\left({t}\right)}{{f}{}\left({t}\right)}$ (7)

References

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