       Re: Area Under Curve (Min Length Interval)

• To: mathgroup at smc.vnet.net
• Subject: [mg53909] Re: [mg53889] Area Under Curve (Min Length Interval)
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Fri, 4 Feb 2005 04:11:03 -0500 (EST)
• Reply-to: hanlonr at cox.net
• Sender: owner-wri-mathgroup at wolfram.com

```The minimum interval requires PDF[dist, a] == PDF[dist, b]

Needs["Statistics`"];
Needs["Graphics`"];

minimumInterval[dist_, area_?NumericQ] :=
Module[{a,b,mu=Mean[dist],ae,be},
ae=Max[mu/4,Domain[dist][[1,1]]];
be=Min[2*mu,Domain[dist][[1,2]]];
{a,b} /. FindRoot[{
PDF[dist,a]==PDF[dist,b],
CDF[dist,b]-CDF[dist,a]==area},
{a,ae},{b,be}]];

dist=ChiSquareDistribution;

{a,b}= minimumInterval[dist,0.93]

{0.37253,10.3441}

FilledPlot[
{0,UnitStep[x-a]-UnitStep[x-b],1}*
PDF[dist,x],{x,
Max[a-3,Domain[dist][[1,1]]],
Min[b+3,Domain[dist][[1,2]]]},
PlotPoints->50];

Bob Hanlon

>
> From: Bruce Colletti <vze269bv at verizon.net>
To: mathgroup at smc.vnet.net
> Date: 2005/02/02 Wed AM 06:25:53 EST
> Subject: [mg53909] [mg53889] Area Under Curve (Min Length Interval)
>
> Re Mathematica 5.1.
>
> How would I compute the minimum length interval over which the area
under f(x) is given?
>
> For instance, as shown below, f(x) is the PDF of a chi-square distributed
random variable whose CDF is F[x].  Seeking the minimum length 93%-
interval [a,b], the code returns "Obtained solution does not satisfy the
following constraints within Tolerance -> 0.001..."  Fiddling with options has
been futile.
>
> Any ideas?  Thankx.
>
> Bruce
>
> F[x_] := CDF[ChiSquareDistribution, x]
>
> Minimize[{b - a, F[b] - F[a] == 0.93, b > a > 0}, {a, b}]
>
> NMinimize[{b - a, F[b] - F[a] == 0.93, b > a > 0}, {a, b}]
>
>
>

```

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