Re: Integral of a bivariate function

• To: mathgroup at smc.vnet.net
• Subject: [mg48759] Re: Integral of a bivariate function
• From: omid_rezayi at hotmail.com (Marc)
• Date: Tue, 15 Jun 2004 02:50:05 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Many thanks for your reply Steve. Do you know if it is possible to get
an upper bound on the approximation error of this method in the
general case? For example if one is only interested in upper
p-quantiles of the distribution for small p.

"Steve Luttrell" <steve_usenet at _removemefirst_luttrell.org.uk> wrote in message news:<cagij5\$i9n\$1 at smc.vnet.net>...
> "Marc" <omid_rezayi at hotmail.com> wrote in message
> > For a given bivariate function I want  to calculate the integral of
> > the function over an arbitrary compact region A, for instance over
> > A={(x,y)| f(x,y)=c} for some constant c. The function is smooth and in
> > my application it is the joint density of two continuous random
> > variables. I wonder if this can be done in Mathematica and in that
> > case how. Otherwise I'd appreciate any pointer to other programs which
> > can be used for this.
> >
>
> Here is a notebook that describes how I would solve this problem. Select
> from the first (*** to the last ****) and copy/paste anywhere in
> Mathematica; it will automatically detect that you are pasting a whole
> notebook.
>
> Steve Luttrell
>
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> Cell[TextData[{
>   "Compare the above with the analytic result that can be computed in \
> this case. Here are the steps in a  quick derivation.\n\n",
>   Cell[BoxData[
>       FormBox[
>         RowBox[{\(Pr \((a)\)\), "=",
>           RowBox[{"\[Integral]",
>             RowBox[{
>               StyleBox[
>                 RowBox[{"d",
>                   StyleBox["x",
>                     FontSlant->"Italic"]}]], " ",
>               StyleBox[
>                 RowBox[{"d",
>                   StyleBox["y",
>                     FontSlant->"Italic"]}]],
>               " ", \(Pr(x, y)\), \(\[Delta](a - f(x, y))\)}]}]}],
>   "\n\n",
>   Cell[BoxData[
>       FormBox[
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>           RowBox[{\(\[Integral]\_0\%\[Infinity]\),
>             RowBox[{\(1\/2\), " ",
>               RowBox[{"d", "(",
>                 SuperscriptBox[
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>                 StyleBox[")",
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>                 SubsuperscriptBox[
>                   StyleBox["\[Integral]",
>                     FontSlant->"Italic"], "0", \(2  \[Pi]\)],
>                 " ", \(d\[Theta]\ \ \(1\/\((\(\@\(2  \[Pi]\)\) \
> \[Sigma])\)\^2\)
>                   Exp[\(-\(r\^2\/\(2  \[Sigma]\^2\)\)\)] \(\[Delta](
>   "\n\n",
>   Cell[BoxData[
>           2  \[Pi] \( 1\/\((\(\@\(2  \[Pi]\)\) \[Sigma])\)\^2\)
>           Exp[\(-\(a\^2\/\(2  \[Sigma]\^2\)\)\)]\)]],
>   "\n\n",
>   Cell[BoxData[
>           Exp[\(-\(a\/\(2  \[Sigma]\^2\)\)\)]\)]]
> }], "Text"],
>
> Cell[TextData[{
>   "Setting ",
>   Cell[BoxData[
>   ", plot this over the same range of ",
>   Cell[BoxData[
>   " as the numerical approximation above."
> }], "Text"],
>
> Cell[CellGroupData[{
>
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>     \(\(g2 =
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>               0.1}]];\)\)], "Input"],
>
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