Mills Ratio

Author

Eric W. Weisstein
October 24, 2024

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Statistics/MillsRatio.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/MillsRatio.html.

©2024 Wolfram Research, Inc. except for portions noted otherwise

Definition

((1-CDF[dist,x])/(PDF[dist,x]))

Examples

Normal Distribution

dist=NormalDistribution[μ,σ];

FullSimplify[((1-CDF[dist,x])/(PDF[dist,x]))]

FullSimplify[((1-CDF[dist,x])/(PDF[dist,x]))/.{μ->0,σ->1}]

ids=(EntityClass["continued fraction identities containing Erfc", EntityClass["ContinuedFraction","ErfcAll"], "EntityClass[\"ContinuedFraction\", \"ErfcAll\"]", "continued fraction identities"])//EntityList

TextGrid[{#,#["ContinuedFractionIdentity"]}&/@ids,Dividers->All]//TraditionalForm

(Entity[{"Erfc",1}, Entity["ContinuedFraction",{"Erfc",1}], "Entity[\"ContinuedFraction\", {\"Erfc\", 1}]", "continued fraction identity"])["ContinuedFractionIdentity"]

1/(x+ContinuedFractionK[k,x,{k,∞}])

ContinuedFractionK[Piecewise[{{1,k==1}},k-1],x,{k,∞}]

ContinuedFractionK[Piecewise[{{1,k==1}},k-1],x,{k,10}]

ContinuedFractionK[Piecewise[{{1,k==1}},k-1],x,{k,20}]==(1/(x+(Underoverscript[\[ContinuedFractionK], (k=1), 19])(k/x)))

Plot[Exp[x^2/2] Sqrt[Pi/2] Erfc[x/Sqrt@2]-1/(x + ContinuedFractionK[k, x, {k, 1, 10^5}]),{x,.1,.2}]

Plot[Exp[x^2/2] Sqrt[Pi/2] Erfc[x/Sqrt@2]-1/(x + ContinuedFractionK[k, x, {k, 1, 10^5}]),{x,0,1},PlotRange->{-.1,.01}]