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]).

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

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

$\mathrm{MillsRatio}\left('\mathrm{{\rm B}}'\left(p\,q\right)\,t\right)$

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

$\mathrm{MillsRatio}\left('\mathrm{{\rm B}}'\left(3\&comma;5\right)\&comma;\frac{1}{2}\right)$

$\frac{64}{105}-\frac{8\mathrm{hypergeom}\left(\left[\mathrm{-4}\&comma;3\right]\&comma;\left[4\right]\&comma;\frac{1}{2}\right)}{3}$

$\mathrm{MillsRatio}\left('\mathrm{{\rm B}}'\left(3\&comma;5\right)\&comma;\frac{1}{2}\&comma;\mathrm{numeric}\right)$

$0.138095238095238$

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

$X≔\mathrm{RandomVariable}\left(T\right)\&colon;$

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

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

$\mathrm{MillsRatio}\left(X\&comma;t\right)$

$\left(1-\frac{\mathrm{\pi }+2\arctan \left(t\right)}{2\mathrm{\pi }}\right)\mathrm{\pi }\left(t^2+1\right)$

$U≔\mathrm{Distribution}\left(\mathrm{`=`}\left(\mathrm{CDF}\&comma;t\mapsto F\left(t\right)\right)\&comma;\mathrm{`=`}\left(\mathrm{PDF}\&comma;t\mapsto f\left(t\right)\right)\right)\&colon;$

$Y≔\mathrm{RandomVariable}\left(U\right)\&colon;$

$\mathrm{CDF}\left(Y\&comma;t\right)$

$F\left(t\right)$

$\mathrm{MillsRatio}\left(Y\&comma;t\right)$

$\frac{1-F\left(t\right)}{f\left(t\right)}$

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