Re: Integral of a bivariate function
- To: mathgroup at smc.vnet.net
- Subject: [mg48753] Re: Integral of a bivariate function
- From: "Steve Luttrell" <steve_usenet at _removemefirst_luttrell.org.uk>
- Date: Sat, 12 Jun 2004 23:33:42 -0400 (EDT)
- References: <cadvmq$6mf$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
"Marc" <omid_rezayi at hotmail.com> wrote in message news:cadvmq$6mf$1 at smc.vnet.net... > 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 (************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info at wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 86575, 2344]*) (*NotebookOutlinePosition[ 87221, 2366]*) (* CellTagsIndexPosition[ 87177, 2362]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Mapping a Probability Density", "Title"], Cell["\<\ Thoughts on how to map a PDF using a Gaussian approximation to the \ Dirac delta function S P Luttrell 12 June 2004\ \>", "Subtitle"], Cell[TextData[{ "The basic relationship for mapping a PDF is\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))\)}]}]}], TraditionalForm]]], "\n\nwhere ", Cell[BoxData[ \(TraditionalForm\`Pr(x, y)\)]], " is the joint PDF in ", Cell[BoxData[ \(TraditionalForm\`x\)]], " and ", Cell[BoxData[ \(TraditionalForm\`y\)]], ", ", Cell[BoxData[ \(TraditionalForm\`f(x, y)\)]], " maps to the variable whose PDF you wish to compute, and ", Cell[BoxData[ \(TraditionalForm\`\[Delta](a - f(x, y))\)]], " is a Dirac delta function that constrains the integral over ", Cell[BoxData[ \(TraditionalForm\`x\)]], " and ", Cell[BoxData[ \(TraditionalForm\`y\)]], " to pick up only those parts of ", Cell[BoxData[ \(TraditionalForm\`Pr(x, y)\)]], " that contribute to ", Cell[BoxData[ \(TraditionalForm\`Pr(a)\)]], "." }], "Text"], Cell[TextData[{ "Define a Gaussian PDF ", Cell[BoxData[ \(TraditionalForm\`Pr(x, y)\)]], " to work with." }], "Text"], Cell[BoxData[ \(\(p[x_, y_, \[Sigma]_] := \(1\/\((\(\@\(2 \[Pi]\)\) \[Sigma])\)\^2\ \) Exp[\(-\(\(x\^2 + y\^2\)\/\(2 \[Sigma]\^2\)\)\)];\)\)], "Input"], Cell["Check that it is correctly normalised.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NIntegrate[ p[x, y, 1], {x, \(-\[Infinity]\), \[Infinity]}, {y, \(-\ \[Infinity]\), \[Infinity]}]\)], "Input"], Cell[BoxData[ \(1.0000000236891413`\)], "Output"] }, Open ]], Cell[TextData[{ "Define an ", Cell[BoxData[ \(TraditionalForm\`f(x, y)\)]], " to work with. Curves of constant ", Cell[BoxData[ \(TraditionalForm\`f(x, y)\)]], " are circles centred on the origin, so the PDF we are going to \ compute is the probability density as a function of squared radius." }], "Text"], Cell[BoxData[ \(\(f[x_, y_] := x\^2 + y\^2;\)\)], "Input"], Cell[TextData[{ "Define an approximation to the Dirac delta function. This is a \ Gaussian with standard devaiation ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\)]], ". As ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\[LongRightArrow]0\)]], " this is exactly a Dirac delta function." }], "Text"], Cell[BoxData[ \(\(delta[ z_, \[Epsilon]_] := \(1\/\(\(\@\(2 \[Pi]\)\) \ \[Epsilon]\)\) Exp[\(-\(z\^2\/\(2 \[Epsilon]\^2\)\)\)];\)\)], "Input"], Cell["\<\ Switch off warning messages that occur when integrating an almost \ singular function. This is a dodgy procedure, so the quality of the \ numerical results must be verified. This is done below.\ \>", "Text"], Cell[BoxData[ \(Off[NIntegrate::"\<slwcon\>"]\)], "Input"], Cell[TextData[{ "For concretness, fix ", Cell[BoxData[ \(TraditionalForm\`\[Sigma] = 1\)]], ". Check how ", Cell[BoxData[ \(TraditionalForm\`Pr(a = 0.1)\)]], " varies with the width ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\)]], " of the approximation to the Dirac delta function. As ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\[LongRightArrow]0\)]], " this tends to a constant, as expected. This gives an idea of what \ size ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\)]], " should be to obtain a good estimate of ", Cell[BoxData[ \(TraditionalForm\`Pr(a)\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Plot[ NIntegrate[ p[x, y, 1] delta[f[x, y] - 0.1, \[Epsilon]], {x, \(-\[Infinity]\), \[Infinity]}, \ {y, \(-\[Infinity]\), \[Infinity]}], {\[Epsilon], 0.01, 1}];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 -0.362295 2.03012 [ [.21429 .03123 -9 -9 ] [.21429 .03123 9 0 ] [.40476 .03123 -9 -9 ] [.40476 .03123 9 0 ] [.59524 .03123 -9 -9 ] [.59524 .03123 9 0 ] [.78571 .03123 -9 -9 ] [.78571 .03123 9 0 ] [.97619 .03123 -3 -9 ] [.97619 .03123 3 0 ] [.01131 .14524 -24 -4.5 ] [.01131 .14524 0 4.5 ] [.01131 .24674 -18 -4.5 ] [.01131 .24674 0 4.5 ] [.01131 .34825 -24 -4.5 ] [.01131 .34825 0 4.5 ] [.01131 .44975 -18 -4.5 ] [.01131 .44975 0 4.5 ] [.01131 .55126 -24 -4.5 ] [.01131 .55126 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g ..25 Mabswid [ ] 0 setdash ..21429 .04373 m ..21429 .04998 L s [(0.2)] .21429 .03123 0 1 Mshowa ..40476 .04373 m ..40476 .04998 L s [(0.4)] .40476 .03123 0 1 Mshowa ..59524 .04373 m ..59524 .04998 L s [(0.6)] .59524 .03123 0 1 Mshowa ..78571 .04373 m ..78571 .04998 L s [(0.8)] .78571 .03123 0 1 Mshowa ..97619 .04373 m ..97619 .04998 L s [(1)] .97619 .03123 0 1 Mshowa ..125 Mabswid ..07143 .04373 m ..07143 .04748 L s ..11905 .04373 m ..11905 .04748 L s ..16667 .04373 m ..16667 .04748 L s ..2619 .04373 m ..2619 .04748 L s ..30952 .04373 m ..30952 .04748 L s ..35714 .04373 m ..35714 .04748 L s ..45238 .04373 m ..45238 .04748 L s ..5 .04373 m ..5 .04748 L s ..54762 .04373 m ..54762 .04748 L s ..64286 .04373 m ..64286 .04748 L s ..69048 .04373 m ..69048 .04748 L s ..7381 .04373 m ..7381 .04748 L s ..83333 .04373 m ..83333 .04748 L s ..88095 .04373 m ..88095 .04748 L s ..92857 .04373 m ..92857 .04748 L s ..25 Mabswid 0 .04373 m 1 .04373 L s ..02381 .14524 m ..03006 .14524 L s [(0.25)] .01131 .14524 1 0 Mshowa ..02381 .24674 m ..03006 .24674 L s [(0.3)] .01131 .24674 1 0 Mshowa ..02381 .34825 m ..03006 .34825 L s [(0.35)] .01131 .34825 1 0 Mshowa ..02381 .44975 m ..03006 .44975 L s [(0.4)] .01131 .44975 1 0 Mshowa ..02381 .55126 m ..03006 .55126 L s [(0.45)] .01131 .55126 1 0 Mshowa ..125 Mabswid ..02381 .06403 m ..02756 .06403 L s ..02381 .08433 m ..02756 .08433 L s ..02381 .10463 m ..02756 .10463 L s ..02381 .12494 m ..02756 .12494 L s ..02381 .16554 m ..02756 .16554 L s ..02381 .18584 m ..02756 .18584 L s ..02381 .20614 m ..02756 .20614 L s ..02381 .22644 m ..02756 .22644 L s ..02381 .26704 m ..02756 .26704 L s ..02381 .28734 m ..02756 .28734 L s ..02381 .30765 m ..02756 .30765 L s ..02381 .32795 m ..02756 .32795 L s ..02381 .36855 m ..02756 .36855 L s ..02381 .38885 m ..02756 .38885 L s ..02381 .40915 m ..02756 .40915 L s ..02381 .42945 m ..02756 .42945 L s ..02381 .47006 m ..02756 .47006 L s ..02381 .49036 m ..02756 .49036 L s ..02381 .51066 m ..02756 .51066 L s ..02381 .53096 m ..02756 .53096 L s ..02381 .02343 m ..02756 .02343 L s ..02381 .00313 m ..02756 .00313 L s ..02381 .57156 m ..02756 .57156 L s ..02381 .59186 m ..02756 .59186 L s ..02381 .61216 m ..02756 .61216 L s ..25 Mabswid ..02381 0 m ..02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath ..5 Mabswid ..03333 .60327 m ..03793 .60329 L ..04222 .60331 L ..04349 .60331 L ..04468 .60332 L ..04576 .60332 L ..04692 .60331 L ..04819 .60329 L ..04883 .60327 L ..04954 .60324 L ..05078 .60315 L ..05195 .603 L ..05316 .60278 L ..05385 .60261 L ..05448 .60242 L ..05565 .60198 L ..05689 .60137 L ..05926 .59974 L ..06141 .59767 L ..06393 .59448 L ..06665 .5901 L ..07158 .5799 L ..08215 .55098 L ..09331 .51586 L ..1133 .45498 L ..15089 .36601 L ..1702 .33198 L ..19091 .30183 L ..23186 .25575 L ..2713 .22253 L ..30925 .19713 L ..34962 .17488 L ..3885 .15673 L ..42587 .14152 L ..46567 .12715 L ..50397 .11471 L ..5447 .10269 L ..58393 .09206 L ..62166 .08256 L ..66181 .07312 L ..70047 .06459 L ..74155 .05604 L ..78113 .04825 L ..81921 .04111 L ..85972 .03386 L ..89873 .02719 L ..93624 .02103 L ..97618 .01472 L ..97619 .01472 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg<PAVmbKF5d0`40004P0000/B000`400?l00000o`00003o o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ 0?ooo`00o`3ooolQ0?ooo`006`3oool00`000000oooo0?ooo`0X0?ooo`800000103oool2 000000<0oooo1000000T0?ooo`800000103oool2000000D0oooo0`00000S0?ooo`800000 103oool2000000<0oooo0`00000T0?ooo`800000103oool2000000@0oooo0P00000Z0?oo o`D000001@3oool001/0oooo0P00000X0?ooo`040000003oool0oooo000000P0oooo00<0 00000?ooo`3oool0903oool010000000oooo0?ooo`00000;0?ooo`030000003oool0oooo 0240oooo00@000000?ooo`3oool00000203oool010000000oooo0?ooo`00000R0?ooo`04 0000003oool0oooo000000P0oooo00@000000?ooo`3oool00000:`3oool00`000000oooo 0?ooo`050?ooo`006`3oool00`000000oooo0?ooo`0W0?ooo`040000003oool0oooo0000 00T0oooo00<000000?ooo`3oool08`3oool010000000oooo0?ooo`0000080?ooo`D00000 8P3oool010000000oooo0?ooo`0000080?ooo`040000003oool0oooo00000280oooo00@0 00000?ooo`3oool00000203oool010000000oooo0?ooo`00000[0?ooo`030000003oool0 oooo00D0oooo000K0?ooo`030000003oool0oooo02L0oooo00@000000?ooo`3oool00000 2P3oool00`000000oooo0?ooo`0R0?ooo`040000003oool0oooo000000P0oooo00@00000 0?ooo`3oool000008`3oool010000000oooo0?ooo`0000080?ooo`<000008`3oool01000 0000oooo0?ooo`0000090?ooo`800000;03oool00`000000oooo0?ooo`050?ooo`006`3o ool00`000000oooo0?ooo`0W0?ooo`040000003oool0oooo000000P0oooo00@000000?oo o`3oool000008`3oool010000000oooo0?ooo`0000090?ooo`030000003oool0000002<0 oooo00@000000?ooo`3oool000002@3oool00`000000oooo0?ooo`0R0?ooo`040000003o ool0oooo000000P0oooo00@000000?ooo`3oool00000:@3oool3000000L0oooo000K0?oo o`030000003oool0oooo02P0oooo0P00000:0?ooo`8000009@3oool2000000/0oooo0P00 000T0?ooo`8000002P3oool3000002<0oooo0P00000:0?ooo`8000009@3oool500000003 0?ooo`000000000000L0oooo000K0?ooo`800000k@3oool7000000l0oooo000K0?ooo`03 0000003oool0oooo0>@0oooo2000000F0?ooo`006`3oool00`000000oooo0?ooo`3N0?oo o`H000007P3oool001/0oooo00<000000?ooo`3oool0f@3oool5000002@0oooo000K0?oo o`030000003oool0oooo0=<0oooo1P00000Y0?ooo`006`3oool00`000000oooo0?ooo`3> 0?ooo`D00000;`3oool001@0ooooo`00000=000000006`3oool00`000000oooo0?ooo`09 0?ooo`030000003oool0oooo00X0oooo00<000000?ooo`3oool02P3oool00`000000oooo 0?ooo`090?ooo`030000003oool0oooo00X0oooo00<000000?ooo`3oool02P3oool00`00 0000oooo0?ooo`090?ooo`030000003oool0oooo00X0oooo00<000000?ooo`3oool02P3o ool00`000000oooo0?ooo`090?ooo`030000003oool0oooo00X0oooo00<000000?ooo`3o ool02P3oool00`000000oooo0?ooo`090?ooo`030000003oool0oooo00X0oooo00<00000 0?ooo`3oool02P3oool00`000000oooo0?ooo`060?ooo`D000002`3oool00`000000oooo 0?ooo`0:0?ooo`030000003oool0oooo00T0oooo00<000000?ooo`3oool02P3oool00`00 0000oooo0?ooo`050?ooo`006`3oool00`000000oooo0?ooo`2o0?ooo`D00000?P3oool0 01/0oooo00<000000?ooo`3oool0^P3oool5000004<0oooo000K0?ooo`030000003oool0 oooo0;@0oooo1P0000180?ooo`006`3oool200000;40oooo1000001>0?ooo`006`3oool0 0`000000oooo0?ooo`2]0?ooo`<00000DP3oool001/0oooo00<000000?ooo`3oool0Z@3o ool4000005D0oooo000K0?ooo`030000003oool0oooo0:D0oooo1000001I0?ooo`006`3o ool00`000000oooo0?ooo`2O0?ooo`H00000G@3oool001/0oooo00<000000?ooo`3oool0 V`3oool4000006<0oooo000K0?ooo`800000V@3oool3000006L0oooo000K0?ooo`030000 003oool0oooo09@0oooo1000001Z0?ooo`006`3oool00`000000oooo0?ooo`2@0?ooo`@0 0000KP3oool001/0oooo00<000000?ooo`3oool0RP3oool600000780oooo000K0?ooo`03 0000003oool0oooo08H0oooo1000001h0?ooo`006`3oool2000008D0oooo0P00001l0?oo o`006`3oool00`000000oooo0?ooo`210?ooo`<00000OP3oool001/0oooo00<000000?oo o`3oool0O`3oool200000840oooo000K0?ooo`030000003oool0oooo07`0oooo0`000023 0?ooo`006`3oool00`000000oooo0?ooo`1h0?ooo`@00000QP3oool001/0oooo0P00001e 0?ooo`@00000RP3oool001/0oooo00<000000?ooo`3oool0L@3oool3000008h0oooo000K 0?ooo`030000003oool0oooo06l0oooo0P00002A0?ooo`006`3oool00`000000oooo0?oo o`1/0?ooo`<00000T`3oool00080oooo0P0000040?ooo`8000000`3oool4000000<0oooo 0P0000050?ooo`030000003oool0oooo06X0oooo0P00002F0?ooo`0000D0oooo0000003o ool0oooo000000080?ooo`030000003oool0oooo00<0oooo00@000000?ooo`3oool00000 103oool00`000000oooo0?ooo`1W0?ooo`<00000V03oool000050?ooo`000000oooo0?oo o`0000002@3oool00`000000oooo0?ooo`050?ooo`030000003oool0oooo0080oooo0P00 001V0?ooo`800000V`3oool000050?ooo`000000oooo0?ooo`0000002P3oool00`000000 oooo0?ooo`020?ooo`8000001@3oool00`000000oooo0?ooo`1R0?ooo`<00000W@3oool0 00050?ooo`000000oooo0?ooo`000000203oool010000000oooo0?ooo`0000030?ooo`03 0000003oool0oooo00@0oooo00<000000?ooo`3oool0H03oool200000:00oooo00020?oo o`8000002P3oool2000000@0oooo0`0000040?ooo`030000003oool0oooo05d0oooo0`00 002R0?ooo`006`3oool00`000000oooo0?ooo`1K0?ooo`800000Y@3oool001/0oooo0P00 001J0?ooo`800000Y`3oool001/0oooo00<000000?ooo`3oool0E`3oool200000:T0oooo 000K0?ooo`030000003oool0oooo05D0oooo0P00002[0?ooo`006`3oool00`000000oooo 0?ooo`1C0?ooo`800000[@3oool001/0oooo00<000000?ooo`3oool0D@3oool200000:l0 oooo000K0?ooo`030000003oool0oooo04l0oooo0P00002a0?ooo`006`3oool2000004l0 oooo00<000000?ooo`3oool0/@3oool001/0oooo00<000000?ooo`3oool0C03oool20000 0;@0oooo000K0?ooo`030000003oool0oooo04X0oooo0P00002f0?ooo`006`3oool00`00 0000oooo0?ooo`190?ooo`030000003oool0oooo0;H0oooo000K0?ooo`030000003oool0 oooo04L0oooo0P00002i0?ooo`006`3oool2000004H0oooo0P00002k0?ooo`006`3oool0 0`000000oooo0?ooo`140?ooo`030000003oool0oooo0;/0oooo000K0?ooo`030000003o ool0oooo0480oooo0P00002n0?ooo`006`3oool00`000000oooo0?ooo`100?ooo`800000 `03oool001/0oooo00<000000?ooo`3oool0?`3oool00`000000oooo0?ooo`300?ooo`00 6`3oool2000003l0oooo00<000000?ooo`3oool0`@3oool001/0oooo00<000000?ooo`3o ool0?03oool200000<@0oooo000K0?ooo`030000003oool0oooo03/0oooo00<000000?oo o`3oool0a03oool001/0oooo00<000000?ooo`3oool0>P3oool00`000000oooo0?ooo`35 0?ooo`00203oool2000000@0oooo0P0000040?ooo`8000001@3oool00`000000oooo0?oo o`0i0?ooo`030000003oool0oooo0<H0oooo00070?ooo`040000003oool0oooo000000P0 oooo00@000000?ooo`3oool00000103oool00`000000oooo0?ooo`0h0?ooo`030000003o ool0oooo0<L0oooo00070?ooo`040000003oool0oooo000000X0oooo00<000000?ooo`3o ool00`3oool2000003L0oooo0P00003:0?ooo`001`3oool010000000oooo0?ooo`00000; 0?ooo`030000003oool0oooo0080oooo00<000000?ooo`3oool0=@3oool00`000000oooo 0?ooo`3:0?ooo`001`3oool010000000oooo0?ooo`0000080?ooo`040000003oool0oooo 000000@0oooo00<000000?ooo`3oool0=03oool00`000000oooo0?ooo`3;0?ooo`00203o ool2000000X0oooo0P0000050?ooo`030000003oool0oooo03<0oooo00<000000?ooo`3o ool0c03oool001/0oooo00<000000?ooo`3oool0<P3oool00`000000oooo0?ooo`3=0?oo o`006`3oool200000380oooo00<000000?ooo`3oool0cP3oool001/0oooo00<000000?oo o`3oool0<03oool00`000000oooo0?ooo`3?0?ooo`006`3oool00`000000oooo0?ooo`0_ 0?ooo`030000003oool0oooo0=00oooo000K0?ooo`030000003oool0oooo02h0oooo00<0 00000?ooo`3oool0d@3oool001/0oooo00<000000?ooo`3oool0;P3oool00`000000oooo 0?ooo`3A0?ooo`006`3oool00`000000oooo0?ooo`0]0?ooo`030000003oool0oooo0=80 oooo000K0?ooo`800000;@3oool00`000000oooo0?ooo`3C0?ooo`006`3oool00`000000 oooo0?ooo`0[0?ooo`030000003oool0oooo0=@0oooo000K0?ooo`030000003oool0oooo 02X0oooo00<000000?ooo`3oool0e@3oool001/0oooo00<000000?ooo`3oool0:@3oool0 0`000000oooo0?ooo`3F0?ooo`006`3oool00`000000oooo0?ooo`0X0?ooo`030000003o ool0oooo0=L0oooo000K0?ooo`800000:@3oool00`000000oooo0?ooo`3G0?ooo`006`3o ool00`000000oooo0?ooo`0W0?ooo`030000003oool0oooo0=P0oooo000K0?ooo`030000 003oool0oooo02H0oooo00<000000?ooo`3oool0f@3oool001/0oooo00<000000?ooo`3o ool09P3oool00`000000oooo0?ooo`3I0?ooo`006`3oool00`000000oooo0?ooo`0U0?oo o`030000003oool0oooo0=X0oooo000K0?ooo`8000009P3oool00`000000oooo0?ooo`3J 0?ooo`006`3oool00`000000oooo0?ooo`0T0?ooo`030000003oool0oooo0=/0oooo000K 0?ooo`030000003oool0oooo02<0oooo00<000000?ooo`3oool0g03oool001/0oooo00<0 00000?ooo`3oool08`3oool00`000000oooo0?ooo`3L0?ooo`000P3oool2000000@0oooo 0P0000040?ooo`800000103oool2000000D0oooo00<000000?ooo`3oool08P3oool00`00 0000oooo0?ooo`3M0?ooo`0000D0oooo0000003oool0oooo000000080?ooo`040000003o ool0oooo00000080oooo00@000000?ooo`3oool00000103oool00`000000oooo0?ooo`0R 0?ooo`030000003oool0oooo0=d0oooo00001@3oool000000?ooo`3oool0000000X0oooo 00<000000?ooo`3oool0103oool00`000000oooo0?ooo`020?ooo`8000008P3oool00`00 0000oooo0?ooo`3N0?ooo`0000D0oooo0000003oool0oooo0000000;0?ooo`040000003o ool0oooo0?ooo`8000001@3oool00`000000oooo0?ooo`0Q0?ooo`030000003oool0oooo 0=h0oooo00001@3oool000000?ooo`3oool0000000P0oooo00@000000?ooo`3oool00000 0`3oool00`000000oooo0?ooo`040?ooo`030000003oool0oooo0200oooo00<000000?oo o`3oool0g`3oool00080oooo0P00000:0?ooo`800000103oool3000000@0oooo00<00000 0?ooo`3oool0803oool00`000000oooo0?ooo`3O0?ooo`006`3oool00`000000oooo0?oo o`0O0?ooo`030000003oool0oooo0>00oooo000K0?ooo`800000803oool00`000000oooo 0?ooo`3P0?ooo`006`3oool00`000000oooo0?ooo`0O0?ooo`030000003oool0oooo0>00 oooo000K0?ooo`030000003oool0oooo01h0oooo00<000000?ooo`3oool0h@3oool001/0 oooo00<000000?ooo`3oool07P3oool00`000000oooo0?ooo`3Q0?ooo`006`3oool00`00 0000oooo0?ooo`0M0?ooo`030000003oool0oooo0>80oooo000K0?ooo`030000003oool0 oooo01d0oooo00<000000?ooo`3oool0hP3oool001/0oooo0P00000M0?ooo`030000003o ool0oooo0><0oooo000K0?ooo`030000003oool0oooo01`0oooo00<000000?ooo`3oool0 h`3oool001/0oooo00<000000?ooo`3oool0703oool00`000000oooo0?ooo`3S0?ooo`00 6`3oool00`000000oooo0?ooo`0K0?ooo`030000003oool0oooo0>@0oooo000K0?ooo`03 0000003oool0oooo01/0oooo00<000000?ooo`3oool0i03oool001/0oooo0P00000K0?oo o`030000003oool0oooo0>D0oooo000K0?ooo`030000003oool0oooo01X0oooo00<00000 0?ooo`3oool0i@3oool001/0oooo00<000000?ooo`3oool06@3oool00`000000oooo0?oo o`3V0?ooo`006`3oool00`000000oooo0?ooo`0I0?ooo`030000003oool0oooo0>H0oooo 000K0?ooo`030000003oool0oooo01P0oooo00<000000?ooo`3oool0i`3oool001/0oooo 0P00000I0?ooo`030000003oool0oooo0>L0oooo000K0?ooo`030000003oool0oooo01P0 oooo00<000000?ooo`3oool0i`3oool001/0oooo00<000000?ooo`3oool05`3oool00`00 0000oooo0?ooo`3X0?ooo`006`3oool00`000000oooo0?ooo`0G0?ooo`030000003oool0 oooo0>P0oooo00080?ooo`800000103oool2000000D0oooo0`0000030?ooo`030000003o ool0oooo01H0oooo00<000000?ooo`3oool0j@3oool000L0oooo00@000000?ooo`3oool0 00002`3oool00`000000oooo0?ooo`020?ooo`030000003oool0oooo01H0oooo00<00000 0?ooo`3oool0j@3oool000L0oooo00@000000?ooo`3oool00000203oool5000000<0oooo 0P00000F0?ooo`030000003oool0oooo0>X0oooo00070?ooo`040000003oool0oooo0000 00P0oooo00@000000?ooo`3oool00000103oool00`000000oooo0?ooo`0E0?ooo`030000 003oool0oooo0>X0oooo00070?ooo`040000003oool0oooo000000T0oooo00<000000?oo o`000000103oool00`000000oooo0?ooo`0E0?ooo`030000003oool0oooo0>X0oooo0008 0?ooo`8000002`3oool2000000@0oooo00<000000?ooo`3oool0503oool00`000000oooo 0?ooo`3[0?ooo`006`3oool00`000000oooo0?ooo`0D0?ooo`030000003oool0oooo0>/0 oooo000K0?ooo`800000503oool00`000000oooo0?ooo`3/0?ooo`006`3oool00`000000 oooo0?ooo`0C0?ooo`030000003oool0oooo0>`0oooo000K0?ooo`030000003oool0oooo 01<0oooo00<000000?ooo`3oool0k03oool001/0oooo00<000000?ooo`3oool04P3oool0 0`000000oooo0?ooo`3]0?ooo`006`3oool00`000000oooo0?ooo`0B0?ooo`030000003o ool0oooo0>d0oooo000K0?ooo`030000003oool0oooo0180oooo00<000000?ooo`3oool0 k@3oool001/0oooo0P00000B0?ooo`030000003oool0oooo0>h0oooo000K0?ooo`030000 003oool0oooo0140oooo00<000000?ooo`3oool0kP3oool001/0oooo00<000000?ooo`3o ool0403oool00`000000oooo0?ooo`3_0?ooo`006`3oool00`000000oooo0?ooo`0@0?oo o`030000003oool0oooo0>l0oooo000K0?ooo`030000003oool0oooo0100oooo00<00000 0?ooo`3oool0k`3oool001/0oooo0P00000@0?ooo`030000003oool0oooo0?00oooo000K 0?ooo`030000003oool0oooo00l0oooo00<000000?ooo`3oool0l03oool001/0oooo00<0 00000?ooo`3oool03`3oool00`000000oooo0?ooo`3`0?ooo`006`3oool00`000000oooo 0?ooo`0>0?ooo`030000003oool0oooo0?40oooo000K0?ooo`030000003oool0oooo00h0 oooo00<000000?ooo`3oool0l@3oool001/0oooo0P00000?0?ooo`030000003oool0oooo 0?40oooo000K0?ooo`030000003oool0oooo00d0oooo00<000000?ooo`3oool0lP3oool0 01/0oooo00<000000?ooo`3oool03@3oool00`000000oooo0?ooo`3b0?ooo`006`3oool0 0`000000oooo0?ooo`0=0?ooo`030000003oool0oooo0?80oooo00020?ooo`800000103o ool2000000D0oooo0`0000020?ooo`8000001@3oool00`000000oooo0?ooo`0=0?ooo`03 0000003oool0oooo0?80oooo00001@3oool000000?ooo`3oool0000000/0oooo00@00000 0?ooo`3oool000000P3oool00`000000oooo0?ooo`020?ooo`030000003oool0oooo00`0 oooo00<000000?ooo`3oool0l`3oool000050?ooo`000000oooo0?ooo`000000203oool5 000000@0oooo00<000000?ooo`3oool00P3oool2000000d0oooo00<000000?ooo`3oool0 l`3oool000050?ooo`000000oooo0?ooo`000000203oool010000000oooo0?ooo`000003 0?ooo`8000001@3oool00`000000oooo0?ooo`0<0?ooo`030000003oool0oooo0?<0oooo 00001@3oool000000?ooo`3oool0000000T0oooo00<000000?ooo`0000000`3oool00`00 0000oooo0?ooo`040?ooo`030000003oool0oooo00/0oooo00<000000?ooo`3oool0m03o ool00080oooo0P00000;0?ooo`8000000`3oool3000000@0oooo00<000000?ooo`3oool0 2`3oool00`000000oooo0?ooo`3d0?ooo`006`3oool00`000000oooo0?ooo`0;0?ooo`03 0000003oool0oooo0?@0oooo000K0?ooo`800000303oool00`000000oooo0?ooo`3d0?oo o`006`3oool00`000000oooo0?ooo`0:0?ooo`030000003oool0oooo0?D0oooo000K0?oo o`030000003oool0oooo00X0oooo00<000000?ooo`3oool0m@3oool001/0oooo00<00000 0?ooo`3oool02@3oool00`000000oooo0?ooo`3f0?ooo`006`3oool00`000000oooo0?oo o`090?ooo`030000003oool0oooo0?H0oooo000K0?ooo`030000003oool0oooo00P0oooo 00<000000?ooo`3oool0m`3oool001/0oooo0P0000090?ooo`030000003oool0oooo0?L0 oooo000K0?ooo`030000003oool0oooo00L0oooo00<000000?ooo`3oool0n03oool001/0 oooo00<000000?ooo`3oool01@3oool200000?/0oooo000K0?ooo`030000003oool00000 00H00000o03oool001/0oooo00<000000?ooo`3oool0o`3oool30?ooo`006`3oool20000 0?l0oooo103oool001/0oooo00<000000?ooo`3oool0o`3oool30?ooo`006`3oool00`00 0000oooo0?ooo`3o0?ooo`<0oooo003o0?ooob40oooo003o0?ooob40oooo003o0?ooob40 oooo003o0?ooob40oooo003o0?ooob40oooo0000\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.107864, 0.16621, \ 0.00394728, 0.00185177}}] }, Open ]], Cell[TextData[{ "Now fix ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon] = 0.01\)]], " because in the above plot shows this to be a good value to use. \ More generally, a better survey would need to be done to pick a good \ ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\)]], ". Now plot ", Cell[BoxData[ \(TraditionalForm\`Pr(a)\)]], " over a range of values of ", Cell[BoxData[ \(TraditionalForm\`a\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(g1 = Plot[NIntegrate[ p[x, y, 1] delta[f[x, y] - a, 0.01], {x, \(-\[Infinity]\), \[Infinity]}, {y, \(-\ \[Infinity]\), \[Infinity]}], {a, 0, 0.1}];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 9.52381 -6.94429 15.3441 [ [.21429 .10149 -12 -9 ] [.21429 .10149 12 0 ] [.40476 .10149 -12 -9 ] [.40476 .10149 12 0 ] [.59524 .10149 -12 -9 ] [.59524 .10149 12 0 ] [.78571 .10149 -12 -9 ] [.78571 .10149 12 0 ] [.97619 .10149 -9 -9 ] [.97619 .10149 9 0 ] [.01131 .26743 -24 -4.5 ] [.01131 .26743 0 4.5 ] [.01131 .42087 -24 -4.5 ] [.01131 .42087 0 4.5 ] [.01131 .57431 -24 -4.5 ] [.01131 .57431 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g ..25 Mabswid [ ] 0 setdash ..21429 .11399 m ..21429 .12024 L s [(0.02)] .21429 .10149 0 1 Mshowa ..40476 .11399 m ..40476 .12024 L s [(0.04)] .40476 .10149 0 1 Mshowa ..59524 .11399 m ..59524 .12024 L s [(0.06)] .59524 .10149 0 1 Mshowa ..78571 .11399 m ..78571 .12024 L s [(0.08)] .78571 .10149 0 1 Mshowa ..97619 .11399 m ..97619 .12024 L s [(0.1)] .97619 .10149 0 1 Mshowa ..125 Mabswid ..07143 .11399 m ..07143 .11774 L s ..11905 .11399 m ..11905 .11774 L s ..16667 .11399 m ..16667 .11774 L s ..2619 .11399 m ..2619 .11774 L s ..30952 .11399 m ..30952 .11774 L s ..35714 .11399 m ..35714 .11774 L s ..45238 .11399 m ..45238 .11774 L s ..5 .11399 m ..5 .11774 L s ..54762 .11399 m ..54762 .11774 L s ..64286 .11399 m ..64286 .11774 L s ..69048 .11399 m ..69048 .11774 L s ..7381 .11399 m ..7381 .11774 L s ..83333 .11399 m ..83333 .11774 L s ..88095 .11399 m ..88095 .11774 L s ..92857 .11399 m ..92857 .11774 L s ..25 Mabswid 0 .11399 m 1 .11399 L s ..02381 .26743 m ..03006 .26743 L s [(0.47)] .01131 .26743 1 0 Mshowa ..02381 .42087 m ..03006 .42087 L s [(0.48)] .01131 .42087 1 0 Mshowa ..02381 .57431 m ..03006 .57431 L s [(0.49)] .01131 .57431 1 0 Mshowa ..125 Mabswid ..02381 .14468 m ..02756 .14468 L s ..02381 .17537 m ..02756 .17537 L s ..02381 .20605 m ..02756 .20605 L s ..02381 .23674 m ..02756 .23674 L s ..02381 .29812 m ..02756 .29812 L s ..02381 .32881 m ..02756 .32881 L s ..02381 .35949 m ..02756 .35949 L s ..02381 .39018 m ..02756 .39018 L s ..02381 .45156 m ..02756 .45156 L s ..02381 .48225 m ..02756 .48225 L s ..02381 .51294 m ..02756 .51294 L s ..02381 .54362 m ..02756 .54362 L s ..02381 .0833 m ..02756 .0833 L s ..02381 .05261 m ..02756 .05261 L s ..02381 .02192 m ..02756 .02192 L s ..02381 .605 m ..02756 .605 L s ..25 Mabswid ..02381 0 m ..02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath ..5 Mabswid ..15343 0 m ..16372 .12652 L ..17461 .23169 L ..18466 .31264 L ..20381 .42984 L ..2139 .47509 L ..22459 .51284 L ..234 .53869 L ..24413 .56006 L ..25368 .57513 L ..26243 .58535 L ..26719 .5897 L ..27227 .59352 L ..27666 .5962 L ..28141 .59853 L ..28433 .59969 L ..28701 .6006 L ..28952 .60132 L ..29228 .60196 L ..29459 .6024 L ..29707 .60276 L ..29844 .60292 L ..29969 .60304 L ..30095 .60314 L ..30213 .60321 L ..30323 .60326 L ..30442 .6033 L ..30573 .60332 L ..30694 .60332 L ..30813 .6033 L ..30943 .60326 L ..31008 .60323 L ..31079 .6032 L ..31205 .60312 L ..31452 .60292 L ..31685 .60268 L ..32122 .6021 L ..33118 .60024 L ..34198 .59755 L ..3829 .58394 L ..4223 .5689 L ..46019 .55406 L ..50053 .53821 L ..53935 .52298 L ..58063 .50681 L ..62039 .49128 L ..65863 .47636 L ..69933 .46052 L ..73851 .44531 L ..78015 .42917 L Mistroke ..82027 .41366 L ..85887 .39876 L ..89993 .38295 L ..93947 .36776 L ..97619 .35367 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg<PAVmbKF5d0`40004P0000/B000`400?l00000o`00003o o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ 0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`006`3oool00`000000oooo0?ooo`0O0?oo o`030000003oool0oooo0>00oooo000K0?ooo`030000003oool0oooo01l0oooo00<00000 0?ooo`3oool0h03oool001/0oooo00<000000?ooo`3oool07`3oool00`000000oooo0?oo o`3P0?ooo`006`3oool00`000000oooo0?ooo`0O0?ooo`030000003oool0oooo0>00oooo 000K0?ooo`030000003oool0oooo01l0oooo00<000000?ooo`3oool0h03oool001/0oooo 0P00000P0?ooo`030000003oool0oooo0>00oooo000K0?ooo`030000003oool0oooo0200 oooo00<000000?ooo`3oool0g`3oool001/0oooo00<000000?ooo`3oool0803oool00`00 0000oooo0?ooo`3O0?ooo`006`3oool00`000000oooo0?ooo`0P0?ooo`030000003oool0 oooo0=l0oooo000K0?ooo`030000003oool0oooo0200oooo00<000000?ooo`3oool0g`3o ool001/0oooo00<000000?ooo`3oool0803oool00`000000oooo0?ooo`3O0?ooo`006`3o ool00`000000oooo0?ooo`0P0?ooo`030000003oool0oooo0=l0oooo000K0?ooo`030000 003oool0oooo0200oooo00<000000?ooo`3oool0g`3oool001/0oooo00<000000?ooo`3o ool0803oool00`000000oooo0?ooo`3O0?ooo`006`3oool200000240oooo00<000000?oo o`3oool0g`3oool001/0oooo00<000000?ooo`3oool0803oool00`000000oooo0?ooo`3O 0?ooo`006`3oool00`000000oooo0?ooo`0P0?ooo`030000003oool0oooo0=l0oooo000K 0?ooo`030000003oool0oooo0240oooo00<000000?ooo`3oool0gP3oool001/0oooo00<0 00000?ooo`3oool08@3oool010000000oooo0?ooo`3oool2000000@0oooo0P0000040?oo o`8000000`3oool4000001d0oooo0P0000040?ooo`800000103oool2000000D0oooo0`00 000L0?ooo`800000103oool2000000@0oooo0P0000030?ooo`<000007P3oool2000000@0 oooo0P0000040?ooo`800000103oool200000280oooo0P0000040?ooo`8000000`3oool5 00000040oooo000K0?ooo`030000003oool0oooo0240oooo00@000000?ooo`3oool00000 0P3oool00`000000oooo0?ooo`060?ooo`040000003oool0oooo00000080oooo00<00000 0?ooo`3oool07@3oool010000000oooo0?ooo`0000080?ooo`040000003oool0oooo0000 00D0oooo00<000000?ooo`3oool06P3oool010000000oooo0?ooo`0000080?ooo`040000 003oool0oooo00000080oooo00@000000?ooo`3oool00000703oool010000000oooo0?oo o`0000080?ooo`040000003oool0oooo00000080oooo00@000000?ooo`3oool00000803o ool010000000oooo0?ooo`00000:0?ooo`030000003oool0oooo0040oooo000K0?ooo`03 0000003oool0oooo0240oooo00@000000?ooo`3oool000000P3oool00`000000oooo0?oo o`060?ooo`040000003oool0oooo000000<0oooo00<000000?ooo`3oool0703oool01000 0000oooo0?ooo`0000080?ooo`040000003oool0oooo00000080oooo1@00000K0?ooo`04 0000003oool0oooo000000P0oooo00@000000?ooo`3oool000000P3oool010000000oooo 0?ooo`00000L0?ooo`040000003oool0oooo000000P0oooo00@000000?ooo`3oool00000 0P3oool010000000oooo0?ooo`00000P0?ooo`040000003oool0oooo000000X0oooo00<0 00000?ooo`3oool00@3oool001/0oooo00<000000?ooo`3oool08@3oool010000000oooo 0?ooo`0000020?ooo`030000003oool0oooo00H0oooo00@000000?ooo`3oool00000103o ool00`000000oooo0?ooo`0K0?ooo`040000003oool0oooo000000P0oooo00@000000?oo o`3oool000000P3oool010000000oooo0?ooo`00000L0?ooo`040000003oool0oooo0000 00P0oooo00@000000?ooo`3oool000000P3oool3000001d0oooo00@000000?ooo`3oool0 0000203oool010000000oooo0?ooo`0000030?ooo`8000008@3oool010000000oooo0?oo o`00000:0?ooo`030000003oool0oooo0040oooo000K0?ooo`8000008P3oool010000000 oooo0?ooo`0000020?ooo`030000003oool0oooo00H0oooo00@000000?ooo`3oool00000 0P3oool010000000oooo0?ooo`00000L0?ooo`040000003oool0oooo000000P0oooo00@0 00000?ooo`3oool000000`3oool00`000000oooo0000000L0?ooo`040000003oool0oooo 000000P0oooo00@000000?ooo`3oool000000`3oool00`000000oooo0?ooo`0L0?ooo`04 0000003oool0oooo000000P0oooo00@000000?ooo`3oool000000P3oool010000000oooo 0?ooo`00000P0?ooo`040000003oool0oooo000000X0oooo00<000000?ooo`3oool00@3o ool001/0oooo00<000000?ooo`3oool08@3oool010000000oooo0?ooo`3oool2000000X0 oooo0P0000040?ooo`8000007P3oool2000000X0oooo0P0000050?ooo`8000007@3oool2 000000X0oooo0P0000040?ooo`<000007@3oool2000000X0oooo0P0000040?ooo`800000 8P3oool2000000X0oooo0P0000030?ooo`006`3oool00`000000oooo0?ooo`0Q0?ooo`03 0000003oool0oooo0=h0oooo000K0?ooo`030000003oool0oooo0240oooo00<000000?oo o`3oool0gP3oool001/0oooo00<000000?ooo`3oool08@3oool00`000000oooo0?ooo`3N 0?ooo`006`3oool00`000000oooo0?ooo`0Q0?ooo`030000003oool0oooo0=h0oooo000K 0?ooo`030000003oool0oooo0280oooo00<000000?ooo`3oool0g@3oool001/0oooo00<0 00000?ooo`3oool08P3oool00`000000oooo0?ooo`3M0?ooo`005@3ooooo000000T00000 0`3oool001/0oooo00<000000?ooo`3oool02@3oool00`000000oooo0?ooo`0:0?ooo`03 0000003oool0oooo00T0oooo00<000000?ooo`3oool02P3oool00`000000oooo0?ooo`09 0?ooo`030000003oool0oooo00X0oooo00<000000?ooo`3oool02P3oool00`000000oooo 0?ooo`090?ooo`030000003oool0oooo00X0oooo00<000000?ooo`3oool02@3oool00`00 0000oooo0?ooo`0:0?ooo`030000003oool0oooo00T0oooo00<000000?ooo`3oool02P3o ool00`000000oooo0?ooo`090?ooo`030000003oool0oooo00X0oooo00<000000?ooo`3o ool02@3oool00`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00T0oooo00<00000 0?ooo`3oool02P3oool00`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00L0oooo 000K0?ooo`030000003oool0oooo0280oooo00<000000?ooo`3oool0g@3oool001/0oooo 00<000000?ooo`3oool08P3oool00`000000oooo0?ooo`3M0?ooo`006`3oool00`000000 oooo0?ooo`0R0?ooo`030000003oool0oooo0=d0oooo000K0?ooo`030000003oool0oooo 0280oooo00<000000?ooo`3oool0g@3oool001/0oooo00<000000?ooo`3oool08P3oool0 0`000000oooo0?ooo`3M0?ooo`006`3oool00`000000oooo0?ooo`0R0?ooo`030000003o ool0oooo0=d0oooo000K0?ooo`8000008`3oool00`000000oooo0?ooo`3M0?ooo`006`3o ool00`000000oooo0?ooo`0R0?ooo`030000003oool0oooo0=d0oooo000K0?ooo`030000 003oool0oooo0280oooo00<000000?ooo`3oool0g@3oool001/0oooo00<000000?ooo`3o ool08`3oool00`000000oooo0?ooo`3L0?ooo`006`3oool00`000000oooo0?ooo`0S0?oo o`030000003oool0oooo0=`0oooo000K0?ooo`030000003oool0oooo02<0oooo00<00000 0?ooo`3oool0g03oool001/0oooo00<000000?ooo`3oool08`3oool00`000000oooo0?oo o`3L0?ooo`006`3oool00`000000oooo0?ooo`0S0?ooo`030000003oool0oooo0=`0oooo 000K0?ooo`800000903oool00`000000oooo0?ooo`3L0?ooo`006`3oool00`000000oooo 0?ooo`0S0?ooo`030000003oool0oooo0=`0oooo000K0?ooo`030000003oool0oooo02<0 oooo00<000000?ooo`3oool0g03oool001/0oooo00<000000?ooo`3oool08`3oool00`00 0000oooo0?ooo`3L0?ooo`006`3oool00`000000oooo0?ooo`0S0?ooo`030000003oool0 oooo0=`0oooo000K0?ooo`030000003oool0oooo02<0oooo00<000000?ooo`3oool0g03o ool001/0oooo00<000000?ooo`3oool08`3oool00`000000oooo0?ooo`3L0?ooo`006`3o ool00`000000oooo0?ooo`0S0?ooo`030000003oool0oooo0=`0oooo000K0?ooo`800000 903oool00`000000oooo0?ooo`3L0?ooo`006`3oool00`000000oooo0?ooo`0T0?ooo`03 0000003oool0oooo0=/0oooo000K0?ooo`030000003oool0oooo02@0oooo00<000000?oo o`3oool0f`3oool001/0oooo00<000000?ooo`3oool0903oool00`000000oooo0?ooo`3K 0?ooo`006`3oool00`000000oooo0?ooo`0T0?ooo`030000003oool0oooo0=/0oooo000K 0?ooo`030000003oool0oooo02@0oooo00<000000?ooo`3oool0f`3oool001/0oooo00<0 00000?ooo`3oool0903oool00`000000oooo0?ooo`3K0?ooo`006`3oool00`000000oooo 0?ooo`0T0?ooo`030000003oool0oooo0=/0oooo000K0?ooo`8000009@3oool00`000000 oooo0?ooo`3K0?ooo`006`3oool00`000000oooo0?ooo`0T0?ooo`030000003oool0oooo 0=/0oooo000K0?ooo`030000003oool0oooo02@0oooo00<000000?ooo`3oool0f`3oool0 01/0oooo00<000000?ooo`3oool09@3oool00`000000oooo0?ooo`3J0?ooo`006`3oool0 0`000000oooo0?ooo`0U0?ooo`030000003oool0oooo0=X0oooo000K0?ooo`030000003o ool0oooo02D0oooo00<000000?ooo`3oool0fP3oool00080oooo0P0000040?ooo`800000 1@3oool3000000<0oooo00<000000?ooo`3oool00`3oool00`000000oooo0?ooo`0U0?oo o`030000003oool0oooo0=X0oooo00001@3oool000000?ooo`3oool0000000/0oooo00<0 00000?ooo`3oool00P3oool00`000000oooo0?ooo`030?ooo`030000003oool0oooo02D0 oooo00<000000?ooo`3oool0fP3oool000050?ooo`000000oooo0?ooo`000000203oool5 000000<0oooo00<000000?ooo`3oool00`3oool2000002H0oooo00<000000?ooo`3oool0 fP3oool000050?ooo`000000oooo0?ooo`000000203oool010000000oooo0?ooo`000005 0?ooo`030000003oool0oooo0080oooo00<000000?ooo`3oool09@3oool00`000000oooo 0?ooo`3J0?ooo`0000D0oooo0000003oool0oooo000000090?ooo`030000003oool00000 0080oooo00@000000?ooo`3oool00000103oool00`000000oooo0?ooo`0V0?ooo`030000 003oool0oooo0=T0oooo00020?ooo`8000002`3oool200000080oooo100000040?ooo`03 0000003oool0oooo02H0oooo00<000000?ooo`3oool0f@3oool001/0oooo00<000000?oo o`3oool09P3oool00`000000oooo0?ooo`3I0?ooo`006`3oool00`000000oooo0?ooo`0V 0?ooo`030000003oool0oooo0=T0oooo000K0?ooo`030000003oool0oooo02H0oooo00<0 00000?ooo`3oool0f@3oool001/0oooo00<000000?ooo`3oool09P3oool00`000000oooo 0?ooo`3I0?ooo`006`3oool2000002L0oooo00<000000?ooo`3oool0f@3oool001/0oooo 00<000000?ooo`3oool09`3oool00`000000oooo0?ooo`3H0?ooo`006`3oool00`000000 oooo0?ooo`0W0?ooo`030000003oool0oooo0=P0oooo000K0?ooo`030000003oool0oooo 02L0oooo00<000000?ooo`3oool0f03oool001/0oooo00<000000?ooo`3oool09`3oool0 0`000000oooo0?ooo`3H0?ooo`006`3oool00`000000oooo0?ooo`0W0?ooo`030000003o ool0oooo0=P0oooo000K0?ooo`030000003oool0oooo02L0oooo00<000000?ooo`3oool0 f03oool001/0oooo00<000000?ooo`3oool09`3oool00`000000oooo0?ooo`3H0?ooo`00 6`3oool2000002T0oooo00<000000?ooo`3oool0e`3oool001/0oooo00<000000?ooo`3o ool0:03oool00`000000oooo0?ooo`3G0?ooo`006`3oool00`000000oooo0?ooo`0X0?oo o`030000003oool0oooo0=L0oooo000K0?ooo`030000003oool0oooo02P0oooo00<00000 0?ooo`3oool0e`3oool001/0oooo00<000000?ooo`3oool0:03oool00`000000oooo0?oo o`3G0?ooo`006`3oool00`000000oooo0?ooo`0X0?ooo`030000003oool0oooo0=L0oooo 000K0?ooo`030000003oool0oooo02T0oooo00<000000?ooo`3oool0eP3oool001/0oooo 00<000000?ooo`3oool0:@3oool00`000000oooo0?ooo`3;0?ooo`8000002@3oool001/0 oooo0P00000Z0?ooo`030000003oool0oooo0<P0oooo0`00000;0?ooo`006`3oool00`00 0000oooo0?ooo`0Y0?ooo`030000003oool0oooo0<@0oooo1000000>0?ooo`006`3oool0 0`000000oooo0?ooo`0Y0?ooo`030000003oool0oooo0<40oooo0`00000B0?ooo`006`3o ool00`000000oooo0?ooo`0Y0?ooo`030000003oool0oooo0;h0oooo0`00000E0?ooo`00 6`3oool00`000000oooo0?ooo`0Z0?ooo`030000003oool0oooo0;/0oooo0P00000H0?oo o`006`3oool00`000000oooo0?ooo`0Z0?ooo`030000003oool0oooo0;P0oooo0`00000J 0?ooo`006`3oool00`000000oooo0?ooo`0Z0?ooo`030000003oool0oooo0;H0oooo0P00 000M0?ooo`006`3oool00`000000oooo0?ooo`0Z0?ooo`030000003oool0oooo0;@0oooo 0P00000O0?ooo`006`3oool2000002/0oooo00<000000?ooo`3oool0/P3oool200000240 oooo000K0?ooo`030000003oool0oooo02X0oooo00<000000?ooo`3oool0/03oool20000 02<0oooo000K0?ooo`030000003oool0oooo02/0oooo00<000000?ooo`3oool0[@3oool2 000002D0oooo000K0?ooo`030000003oool0oooo02/0oooo00<000000?ooo`3oool0ZP3o ool3000002L0oooo000K0?ooo`030000003oool0oooo02/0oooo00<000000?ooo`3oool0 Y`3oool3000002X0oooo000K0?ooo`030000003oool0oooo02/0oooo00<000000?ooo`3o ool0Y@3oool2000002d0oooo00020?ooo`800000103oool2000000D0oooo0`0000020?oo o`8000001@3oool00`000000oooo0?ooo`0[0?ooo`030000003oool0oooo0:80oooo0`00 000_0?ooo`0000D0oooo0000003oool0oooo0000000;0?ooo`040000003oool0oooo0000 0080oooo00<000000?ooo`3oool00P3oool00`000000oooo0?ooo`0[0?ooo`030000003o ool0oooo09l0oooo0`00000b0?ooo`0000D0oooo0000003oool0oooo000000080?ooo`D0 000000D0oooo0000003oool0oooo000000040?ooo`800000;@3oool00`000000oooo0?oo o`2L0?ooo`800000=@3oool000050?ooo`000000oooo0?ooo`000000203oool010000000 oooo0?ooo`0000030?ooo`8000001@3oool00`000000oooo0?ooo`0/0?ooo`030000003o ool0oooo09T0oooo0`00000g0?ooo`0000D0oooo0000003oool0oooo000000090?ooo`03 0000003oool000000080oooo00@000000?ooo`3oool00000103oool00`000000oooo0?oo o`0/0?ooo`030000003oool0oooo09L0oooo0P00000j0?ooo`000P3oool2000000/0oooo 0P0000030?ooo`8000001@3oool00`000000oooo0?ooo`0/0?ooo`030000003oool0oooo 09@0oooo0`00000l0?ooo`006`3oool00`000000oooo0?ooo`0/0?ooo`030000003oool0 oooo0940oooo0`00000o0?ooo`006`3oool00`000000oooo0?ooo`0/0?ooo`030000003o ool0oooo08l0oooo0P0000120?ooo`006`3oool00`000000oooo0?ooo`0]0?ooo`030000 003oool0oooo08/0oooo0`0000140?ooo`006`3oool00`000000oooo0?ooo`0]0?ooo`03 0000003oool0oooo08P0oooo0`0000170?ooo`006`3oool00`000000oooo0?ooo`0]0?oo o`030000003oool0oooo08H0oooo0P00001:0?ooo`006`3oool2000002h0oooo00<00000 0?ooo`3oool0P`3oool3000004`0oooo000K0?ooo`030000003oool0oooo02h0oooo00<0 00000?ooo`3oool0P03oool2000004l0oooo000K0?ooo`030000003oool0oooo02h0oooo 00<000000?ooo`3oool0O@3oool300000540oooo000K0?ooo`030000003oool0oooo02h0 oooo00<000000?ooo`3oool0NP3oool3000005@0oooo000K0?ooo`030000003oool0oooo 02h0oooo00<000000?ooo`3oool0N03oool2000005L0oooo000K0?ooo`030000003oool0 oooo02l0oooo00<000000?ooo`3oool0M03oool3000005T0oooo000K0?ooo`030000003o ool0oooo02l0oooo00<000000?ooo`3oool0L@3oool3000005`0oooo000K0?ooo`030000 003oool0oooo02l0oooo00<000000?ooo`3oool0K`3oool2000005l0oooo000K0?ooo`80 0000<@3oool00`000000oooo0?ooo`1[0?ooo`<00000H@3oool001/0oooo00<000000?oo o`3oool0<03oool00`000000oooo0?ooo`1Y0?ooo`800000I03oool001/0oooo00<00000 0?ooo`3oool0<03oool00`000000oooo0?ooo`1V0?ooo`<00000IP3oool001/0oooo00<0 00000?ooo`3oool0<03oool00`000000oooo0?ooo`1S0?ooo`<00000J@3oool001/0oooo 00<000000?ooo`3oool0<@3oool00`000000oooo0?ooo`1P0?ooo`800000K03oool001/0 oooo00<000000?ooo`3oool0<@3oool00`000000oooo0?ooo`1M0?ooo`<00000KP3oool0 01/0oooo00<000000?ooo`3oool0<@3oool00`000000oooo0?ooo`1J0?ooo`<00000L@3o ool001/0oooo00<000000?ooo`3oool0<P3oool00`000000oooo0?ooo`1G0?ooo`800000 M03oool001/0oooo0P00000c0?ooo`030000003oool0oooo05@0oooo0`00001f0?ooo`00 6`3oool00`000000oooo0?ooo`0b0?ooo`030000003oool0oooo0580oooo0P00001i0?oo o`006`3oool00`000000oooo0?ooo`0c0?ooo`030000003oool0oooo04h0oooo0`00001k 0?ooo`006`3oool00`000000oooo0?ooo`0c0?ooo`030000003oool0oooo04/0oooo0`00 001n0?ooo`006`3oool00`000000oooo0?ooo`0c0?ooo`030000003oool0oooo04T0oooo 0P0000210?ooo`006`3oool00`000000oooo0?ooo`0d0?ooo`030000003oool0oooo04D0 oooo0`0000230?ooo`006`3oool00`000000oooo0?ooo`0d0?ooo`030000003oool0oooo 04<0oooo0P0000260?ooo`006`3oool00`000000oooo0?ooo`0d0?ooo`030000003oool0 oooo0440oooo0P0000280?ooo`006`3oool2000003H0oooo00<000000?ooo`3oool0?P3o ool2000008X0oooo000K0?ooo`030000003oool0oooo03D0oooo00<000000?ooo`3oool0 ?03oool2000008`0oooo000K0?ooo`030000003oool0oooo03H0oooo00<000000?ooo`3o ool0>@3oool2000008h0oooo000K0?ooo`030000003oool0oooo03H0oooo00<000000?oo o`3oool0=P3oool300000900oooo000K0?ooo`030000003oool0oooo03L0oooo00<00000 0?ooo`3oool0<P3oool3000009<0oooo000K0?ooo`030000003oool0oooo03L0oooo00<0 00000?ooo`3oool0;P3oool4000009H0oooo00020?ooo`800000103oool2000000D0oooo 0`000000103oool000000000000000050?ooo`030000003oool0oooo03P0oooo00<00000 0?ooo`3oool0:P3oool3000009X0oooo00001@3oool000000?ooo`3oool0000000/0oooo 00<000000?ooo`3oool00P3oool00`000000oooo0?ooo`030?ooo`030000003oool0oooo 03P0oooo00<000000?ooo`3oool09`3oool3000009d0oooo00001@3oool000000?ooo`3o ool0000000P0oooo1@0000020?ooo`<00000103oool2000003X0oooo00<000000?ooo`3o ool0903oool200000:00oooo00001@3oool000000?ooo`3oool0000000P0oooo00@00000 0?ooo`3oool000000P3oool010000000oooo0?ooo`0000040?ooo`030000003oool0oooo 03X0oooo00<000000?ooo`3oool0803oool300000:80oooo00001@3oool000000?ooo`3o ool0000000T0oooo00<000000?ooo`0000000P3oool010000000oooo0?ooo`0000040?oo o`030000003oool0oooo03/0oooo00<000000?ooo`3oool0703oool300000:D0oooo0002 0?ooo`8000002`3oool2000000<0oooo0`0000040?ooo`030000003oool0oooo03`0oooo 00<000000?ooo`3oool06@3oool200000:P0oooo000K0?ooo`030000003oool0oooo03d0 oooo00<000000?ooo`3oool05@3oool300000:X0oooo000K0?ooo`030000003oool0oooo 03h0oooo00<000000?ooo`3oool04P3oool200000:d0oooo000K0?ooo`030000003oool0 oooo03l0oooo0`00000>0?ooo`<00000[`3oool001/0oooo00<000000?ooo`3oool0@@3o ool?00000;80oooo000K0?ooo`800000B03oool00`000000oooo0?ooo`2h0?ooo`006`3o ool00`000000oooo0?ooo`3o0?ooo`<0oooo000K0?ooo`030000003oool0oooo0?l0oooo 0`3oool001/0oooo00<000000?ooo`3oool0o`3oool30?ooo`00o`3ooolQ0?ooo`00o`3o oolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?oo o`00o`3ooolQ0?ooo`00\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.0108859, \ 0.450807, 0.000398872, 0.000247573}}] }, Open ]], 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))\)}]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{\(Pr(a)\), "=", RowBox[{\(\[Integral]\_0\%\[Infinity]\), RowBox[{\(1\/2\), " ", RowBox[{"d", "(", SuperscriptBox[ StyleBox["r", FontSlant->"Italic"], "2"], StyleBox[")", FontSlant->"Italic"]}], RowBox[{ SubsuperscriptBox[ StyleBox["\[Integral]", FontSlant->"Italic"], "0", \(2 \[Pi]\)], " ", \(d\[Theta]\ \ \(1\/\((\(\@\(2 \[Pi]\)\) \ \[Sigma])\)\^2\) Exp[\(-\(r\^2\/\(2 \[Sigma]\^2\)\)\)] \(\[Delta]( a - r\^2)\)\)}]}]}]}], TraditionalForm]]], "\n\n", Cell[BoxData[ \(TraditionalForm\`Pr(a) = \(1\/2\) 2 \[Pi] \( 1\/\((\(\@\(2 \[Pi]\)\) \[Sigma])\)\^2\) Exp[\(-\(a\^2\/\(2 \[Sigma]\^2\)\)\)]\)]], "\n\n", Cell[BoxData[ \(TraditionalForm\`Pr(a) = \(1\/\(2 \[Sigma]\^2\)\) Exp[\(-\(a\/\(2 \[Sigma]\^2\)\)\)]\)]] }], "Text"], Cell[TextData[{ "Setting ", Cell[BoxData[ \(TraditionalForm\`\[Sigma] = 1\)]], ", plot this over the same range of ", Cell[BoxData[ \(TraditionalForm\`a\)]], " as the numerical approximation above." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(g2 = With[{\[Sigma] = 1}, Plot[\(1\/\(2 \[Sigma]\^2\)\) Exp[\(-\(a\/\(2 \[Sigma]\^2\)\)\)], {a, 0, 0.1}]];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 9.52381 -11.4655 24.1377 [ [.21429 .59082 -12 -9 ] [.21429 .59082 12 0 ] [.40476 .59082 -12 -9 ] [.40476 .59082 12 0 ] [.59524 .59082 -12 -9 ] [.59524 .59082 12 0 ] [.78571 .59082 -12 -9 ] [.78571 .59082 12 0 ] [.97619 .59082 -9 -9 ] [.97619 .59082 9 0 ] [.01131 .12057 -24 -4.5 ] [.01131 .12057 0 4.5 ] [.01131 .24125 -30 -4.5 ] [.01131 .24125 0 4.5 ] [.01131 .36194 -24 -4.5 ] [.01131 .36194 0 4.5 ] [.01131 .48263 -30 -4.5 ] [.01131 .48263 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g ..25 Mabswid [ ] 0 setdash ..21429 .60332 m ..21429 .60957 L s [(0.02)] .21429 .59082 0 1 Mshowa ..40476 .60332 m ..40476 .60957 L s [(0.04)] .40476 .59082 0 1 Mshowa ..59524 .60332 m ..59524 .60957 L s [(0.06)] .59524 .59082 0 1 Mshowa ..78571 .60332 m ..78571 .60957 L s [(0.08)] .78571 .59082 0 1 Mshowa ..97619 .60332 m ..97619 .60957 L s [(0.1)] .97619 .59082 0 1 Mshowa ..125 Mabswid ..07143 .60332 m ..07143 .60707 L s ..11905 .60332 m ..11905 .60707 L s ..16667 .60332 m ..16667 .60707 L s ..2619 .60332 m ..2619 .60707 L s ..30952 .60332 m ..30952 .60707 L s ..35714 .60332 m ..35714 .60707 L s ..45238 .60332 m ..45238 .60707 L s ..5 .60332 m ..5 .60707 L s ..54762 .60332 m ..54762 .60707 L s ..64286 .60332 m ..64286 .60707 L s ..69048 .60332 m ..69048 .60707 L s ..7381 .60332 m ..7381 .60707 L s ..83333 .60332 m ..83333 .60707 L s ..88095 .60332 m ..88095 .60707 L s ..92857 .60332 m ..92857 .60707 L s ..25 Mabswid 0 .60332 m 1 .60332 L s ..02381 .12057 m ..03006 .12057 L s [(0.48)] .01131 .12057 1 0 Mshowa ..02381 .24125 m ..03006 .24125 L s [(0.485)] .01131 .24125 1 0 Mshowa ..02381 .36194 m ..03006 .36194 L s [(0.49)] .01131 .36194 1 0 Mshowa ..02381 .48263 m ..03006 .48263 L s [(0.495)] .01131 .48263 1 0 Mshowa ..125 Mabswid ..02381 .02402 m ..02756 .02402 L s ..02381 .04815 m ..02756 .04815 L s ..02381 .07229 m ..02756 .07229 L s ..02381 .09643 m ..02756 .09643 L s ..02381 .1447 m ..02756 .1447 L s ..02381 .16884 m ..02756 .16884 L s ..02381 .19298 m ..02756 .19298 L s ..02381 .21712 m ..02756 .21712 L s ..02381 .26539 m ..02756 .26539 L s ..02381 .28953 m ..02756 .28953 L s ..02381 .31367 m ..02756 .31367 L s ..02381 .3378 m ..02756 .3378 L s ..02381 .38608 m ..02756 .38608 L s ..02381 .41022 m ..02756 .41022 L s ..02381 .43436 m ..02756 .43436 L s ..02381 .45849 m ..02756 .45849 L s ..02381 .50677 m ..02756 .50677 L s ..02381 .53091 m ..02756 .53091 L s ..02381 .55504 m ..02756 .55504 L s ..02381 .57918 m ..02756 .57918 L s ..25 Mabswid ..02381 0 m ..02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath ..5 Mabswid ..02381 .60332 m ..06244 .57886 L ..10458 .55225 L ..14415 .52731 L ..18221 .50337 L ..22272 .47794 L ..26171 .45352 L ..30316 .42761 L ..34309 .40271 L ..3815 .3788 L ..42237 .35341 L ..46172 .32902 L ..49955 .30562 L ..53984 .28074 L ..57861 .25686 L ..61984 .23152 L ..65954 .20716 L ..69774 .18378 L ..73838 .15894 L ..77751 .13509 L ..81909 .10979 L ..85916 .08547 L ..89771 .06211 L ..93871 .03732 L ..97619 .01472 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg<PAVmbKF5d0`40004P0000/B000`400?l00000o`00003o o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ 0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00 8@3oool00`000000oooo0?ooo`3l0?ooo`008@3oool00`000000oooo0?ooo`3l0?ooo`00 8@3oool00`000000oooo0?ooo`3l0?ooo`008@3oool00`000000oooo0?ooo`3b0?ooo`03 0000003oool0oooo00L0oooo000Q0?ooo`030000003oool0oooo0?00oooo0P00000:0?oo o`008@3oool00`000000oooo0?ooo`3^0?ooo`800000303oool00240oooo0P00003^0?oo o`030000003oool0oooo00`0oooo000Q0?ooo`030000003oool0oooo0>/0oooo0P00000? 0?ooo`008@3oool00`000000oooo0?ooo`3Y0?ooo`8000004@3oool00240oooo00<00000 0?ooo`3oool0j03oool00`000000oooo0?ooo`0A0?ooo`008@3oool00`000000oooo0?oo o`3V0?ooo`800000503oool00240oooo00<000000?ooo`3oool0i@3oool00`000000oooo 0?ooo`0D0?ooo`008@3oool200000>@0oooo0P00000G0?ooo`008@3oool00`000000oooo 0?ooo`3Q0?ooo`8000006@3oool00240oooo00<000000?ooo`3oool0h03oool00`000000 oooo0?ooo`0I0?ooo`008@3oool00`000000oooo0?ooo`3N0?ooo`800000703oool00240 oooo00<000000?ooo`3oool0g@3oool00`000000oooo0?ooo`0L0?ooo`008@3oool00`00 0000oooo0?ooo`3K0?ooo`8000007`3oool00240oooo0P00003J0?ooo`8000008@3oool0 0240oooo00<000000?ooo`3oool0f03oool00`000000oooo0?ooo`0Q0?ooo`008@3oool0 0`000000oooo0?ooo`3F0?ooo`800000903oool00240oooo00<000000?ooo`3oool0e03o ool2000002H0oooo000Q0?ooo`030000003oool0oooo0=<0oooo00<000000?ooo`3oool0 9P3oool00240oooo00<000000?ooo`3oool0d@3oool2000002T0oooo000Q0?ooo`800000 d03oool2000002/0oooo000Q0?ooo`030000003oool0oooo0<h0oooo00<000000?ooo`3o ool0:`3oool00240oooo00<000000?ooo`3oool0c03oool2000002h0oooo000Q0?ooo`03 0000003oool0oooo0<X0oooo0P00000`0?ooo`008@3oool00`000000oooo0?ooo`390?oo o`030000003oool0oooo0300oooo00080?ooo`800000103oool2000000D0oooo0`000002 0?ooo`8000001@3oool00`000000oooo0?ooo`370?ooo`800000<`3oool000L0oooo00@0 00000?ooo`3oool000002`3oool010000000oooo0?ooo`0000020?ooo`030000003oool0 oooo0080oooo00<000000?ooo`3oool0a@3oool2000003D0oooo00070?ooo`040000003o ool0oooo000000P0oooo1@0000001@3oool000000?ooo`3oool0000000@0oooo0P000034 0?ooo`800000=`3oool000L0oooo00@000000?ooo`3oool00000203oool010000000oooo 0?ooo`0000030?ooo`8000001@3oool00`000000oooo0?ooo`310?ooo`800000>@3oool0 00L0oooo00@000000?ooo`3oool000002@3oool00`000000oooo000000020?ooo`040000 003oool0oooo000000@0oooo00<000000?ooo`3oool0_`3oool2000003/0oooo00080?oo o`8000002`3oool2000000<0oooo0P0000050?ooo`030000003oool0oooo0;h0oooo00<0 00000?ooo`3oool0>`3oool00240oooo00<000000?ooo`3oool0_03oool2000003h0oooo 000Q0?ooo`030000003oool0oooo0;X0oooo0P0000100?ooo`008@3oool200000;X0oooo 00<000000?ooo`3oool0@03oool00240oooo00<000000?ooo`3oool0]`3oool2000004<0 oooo000Q0?ooo`030000003oool0oooo0;D0oooo0P0000150?ooo`008@3oool00`000000 oooo0?ooo`2d0?ooo`030000003oool0oooo04D0oooo000Q0?ooo`030000003oool0oooo 0;80oooo0P0000180?ooo`008@3oool00`000000oooo0?ooo`2a0?ooo`030000003oool0 oooo04P0oooo000Q0?ooo`800000/@3oool00`000000oooo0?ooo`190?ooo`008@3oool0 0`000000oooo0?ooo`2^0?ooo`800000C03oool00240oooo00<000000?ooo`3oool0[@3o ool00`000000oooo0?ooo`1<0?ooo`008@3oool00`000000oooo0?ooo`2[0?ooo`800000 C`3oool00240oooo00<000000?ooo`3oool0ZP3oool00`000000oooo0?ooo`1?0?ooo`00 8@3oool00`000000oooo0?ooo`2X0?ooo`800000DP3oool00240oooo0P00002W0?ooo`80 0000E03oool00240oooo00<000000?ooo`3oool0Y@3oool00`000000oooo0?ooo`1D0?oo o`008@3oool00`000000oooo0?ooo`2S0?ooo`800000E`3oool00240oooo00<000000?oo o`3oool0X@3oool2000005T0oooo000Q0?ooo`030000003oool0oooo0:00oooo00<00000 0?ooo`3oool0F@3oool00240oooo00<000000?ooo`3oool0WP3oool2000005`0oooo000Q 0?ooo`800000W@3oool2000005h0oooo000Q0?ooo`030000003oool0oooo09/0oooo00<0 00000?ooo`3oool0GP3oool00240oooo00<000000?ooo`3oool0V@3oool200000640oooo 000Q0?ooo`030000003oool0oooo09L0oooo0P00001S0?ooo`008@3oool00`000000oooo 0?ooo`2F0?ooo`030000003oool0oooo06<0oooo00020?ooo`800000103oool2000000D0 oooo0`0000020?ooo`800000103oool2000000D0oooo00<000000?ooo`3oool0U03oool2 000006H0oooo00001@3oool000000?ooo`3oool0000000/0oooo00@000000?ooo`3oool0 00000P3oool010000000oooo0?ooo`0000020?ooo`030000003oool0oooo0080oooo00<0 00000?ooo`3oool0T`3oool00`000000oooo0?ooo`1V0?ooo`0000D0oooo0000003oool0 oooo000000080?ooo`D0000000D0oooo0000003oool0oooo000000050?ooo`030000003o ool0oooo0080oooo0P00002B0?ooo`800000J@3oool000050?ooo`000000oooo0?ooo`00 0000203oool010000000oooo0?ooo`0000030?ooo`800000103oool2000000D0oooo00<0 00000?ooo`3oool0S`3oool2000006/0oooo00001@3oool000000?ooo`3oool0000000T0 oooo00<000000?ooo`0000000P3oool010000000oooo0?ooo`0000030?ooo`030000003o ool0oooo00@0oooo00<000000?ooo`3oool0SP3oool00`000000oooo0?ooo`1[0?ooo`00 0P3oool2000000/0oooo0P0000030?ooo`800000103oool3000000@0oooo00<000000?oo o`3oool0S03oool2000006h0oooo000Q0?ooo`030000003oool0oooo08/0oooo00<00000 0?ooo`3oool0KP3oool00240oooo00<000000?ooo`3oool0R@3oool200000740oooo000Q 0?ooo`800000R03oool2000007<0oooo000Q0?ooo`030000003oool0oooo08H0oooo00<0 00000?ooo`3oool0L`3oool00240oooo00<000000?ooo`3oool0Q03oool2000007H0oooo 000Q0?ooo`030000003oool0oooo0880oooo0P00001h0?ooo`008@3oool00`000000oooo 0?ooo`210?ooo`030000003oool0oooo07P0oooo000Q0?ooo`030000003oool0oooo07l0 oooo0P00001k0?ooo`008@3oool2000007h0oooo0P00001m0?ooo`008@3oool00`000000 oooo0?ooo`1l0?ooo`030000003oool0oooo07d0oooo000Q0?ooo`030000003oool0oooo 07X0oooo0P0000200?ooo`008@3oool00`000000oooo0?ooo`1h0?ooo`800000PP3oool0 0240oooo00<000000?ooo`3oool0M`3oool00`000000oooo0?ooo`220?ooo`008@3oool0 0`000000oooo0?ooo`1e0?ooo`800000Q@3oool00240oooo0P00001d0?ooo`800000Q`3o ool00240oooo00<000000?ooo`3oool0LP3oool00`000000oooo0?ooo`270?ooo`008@3o ool00`000000oooo0?ooo`1`0?ooo`800000RP3oool00240oooo00<000000?ooo`3oool0 KP3oool2000008`0oooo000Q0?ooo`030000003oool0oooo06d0oooo00<000000?ooo`3o ool0S03oool00240oooo00<000000?ooo`3oool0J`3oool2000008l0oooo000Q0?ooo`80 0000JP3oool200000940oooo000Q0?ooo`030000003oool0oooo06P0oooo00<000000?oo o`3oool0T@3oool00240oooo00<000000?ooo`3oool0IP3oool2000009@0oooo000Q0?oo o`030000003oool0oooo06@0oooo0P00002F0?ooo`008@3oool00`000000oooo0?ooo`1S 0?ooo`030000003oool0oooo09H0oooo00080?ooo`800000103oool2000000D0oooo0`00 0000103oool000000000000000050?ooo`030000003oool0oooo0640oooo0P00002I0?oo o`001`3oool010000000oooo0?ooo`00000;0?ooo`030000003oool0oooo0080oooo00<0 00000?ooo`3oool00`3oool00`000000oooo0?ooo`1P0?ooo`030000003oool0oooo09T0 oooo00070?ooo`040000003oool0oooo000000P0oooo1@0000020?ooo`<00000103oool2 00000600oooo00<000000?ooo`3oool0VP3oool000L0oooo00@000000?ooo`3oool00000 203oool010000000oooo0?ooo`0000020?ooo`040000003oool0oooo000000@0oooo00<0 00000?ooo`3oool0G@3oool2000009d0oooo00070?ooo`040000003oool0oooo000000T0 oooo00<000000?ooo`0000000P3oool010000000oooo0?ooo`0000040?ooo`030000003o ool0oooo05`0oooo00<000000?ooo`3oool0W@3oool000P0oooo0P00000;0?ooo`800000 0`3oool3000000@0oooo00<000000?ooo`3oool0FP3oool200000:00oooo000Q0?ooo`03 0000003oool0oooo05T0oooo00<000000?ooo`3oool0X03oool00240oooo00<000000?oo o`3oool0E`3oool200000:<0oooo000Q0?ooo`800000EP3oool200000:D0oooo000Q0?oo o`030000003oool0oooo05@0oooo00<000000?ooo`3oool0Y@3oool00240oooo00<00000 0?ooo`3oool0DP3oool200000:P0oooo000Q0?ooo`030000003oool0oooo0500oooo0P00 002Z0?ooo`008@3oool00`000000oooo0?ooo`1?0?ooo`030000003oool0oooo0:X0oooo 000Q0?ooo`030000003oool0oooo04d0oooo0P00002]0?ooo`008@3oool2000004d0oooo 00<000000?ooo`3oool0[@3oool00240oooo00<000000?ooo`3oool0B`3oool00`000000 oooo0?ooo`2^0?ooo`008@3oool00`000000oooo0?ooo`190?ooo`800000/@3oool00240 oooo00<000000?ooo`3oool0B03oool00`000000oooo0?ooo`2a0?ooo`008@3oool00`00 0000oooo0?ooo`160?ooo`800000]03oool00240oooo00<000000?ooo`3oool0A@3oool0 0`000000oooo0?ooo`2d0?ooo`008@3oool2000004@0oooo0P00002g0?ooo`008@3oool0 0`000000oooo0?ooo`110?ooo`800000^@3oool00240oooo00<000000?ooo`3oool0?`3o ool200000;/0oooo000Q0?ooo`030000003oool0oooo03d0oooo0P00002m0?ooo`008@3o ool00`000000oooo0?ooo`0k0?ooo`800000_`3oool00240oooo00<000000?ooo`3oool0 >P3oool00`000000oooo0?ooo`2o0?ooo`008@3oool2000003T0oooo0P0000320?ooo`00 8@3oool00`000000oooo0?ooo`0f0?ooo`800000a03oool00240oooo00<000000?ooo`3o ool0=@3oool00`000000oooo0?ooo`340?ooo`008@3oool00`000000oooo0?ooo`0c0?oo o`800000a`3oool00240oooo00<000000?ooo`3oool0<@3oool200000<T0oooo00020?oo o`800000103oool2000000D0oooo0`000000103oool000000000000000040?ooo`800000 1@3oool00`000000oooo0?ooo`0`0?ooo`030000003oool0oooo0<T0oooo00001@3oool0 00000?ooo`3oool0000000/0oooo00<000000?ooo`3oool00P3oool01@000000oooo0?oo o`3oool000000080oooo00<000000?ooo`3oool00P3oool00`000000oooo0?ooo`0^0?oo o`800000c03oool000050?ooo`000000oooo0?ooo`000000203oool500000080oooo0`00 00050?ooo`030000003oool0oooo0080oooo0P00000^0?ooo`030000003oool0oooo0<`0 oooo00001@3oool000000?ooo`3oool0000000P0oooo00@000000?ooo`3oool000000P3o ool010000000oooo0?ooo`0000030?ooo`8000001@3oool00`000000oooo0?ooo`0[0?oo o`800000c`3oool000050?ooo`000000oooo0?ooo`0000002@3oool00`000000oooo0000 00020?ooo`040000003oool0oooo000000<0oooo00<000000?ooo`3oool0103oool00`00 0000oooo0?ooo`0Y0?ooo`800000d@3oool00080oooo0P00000;0?ooo`8000000`3oool3 000000<0oooo0`0000040?ooo`030000003oool0oooo02P0oooo00<000000?ooo`3oool0 d@3oool00240oooo00<000000?ooo`3oool09P3oool200000=@0oooo000Q0?ooo`030000 003oool0oooo02D0oooo00<000000?ooo`3oool0e03oool00240oooo0P00000T0?ooo`80 0000e`3oool00240oooo00<000000?ooo`3oool08P3oool00`000000oooo0?ooo`3G0?oo o`008@3oool00`000000oooo0?ooo`0P0?ooo`800000fP3oool00240oooo00<000000?oo o`3oool07`3oool00`000000oooo0?ooo`3J0?ooo`008@3oool00`000000oooo0?ooo`0M 0?ooo`800000g@3oool00240oooo00<000000?ooo`3oool0703oool00`000000oooo0?oo o`3M0?ooo`008@3oool2000001/0oooo0P00003P0?ooo`008@3oool00`000000oooo0?oo o`0I0?ooo`030000003oool0oooo0>00oooo000Q0?ooo`030000003oool0oooo01L0oooo 0P00003S0?ooo`008@3oool00`000000oooo0?ooo`0E0?ooo`800000i@3oool00240oooo 00<000000?ooo`3oool0503oool00`000000oooo0?ooo`3U0?ooo`008@3oool00`000000 oooo0?ooo`0B0?ooo`800000j03oool00240oooo0P00000B0?ooo`030000003oool0oooo 0>P0oooo000Q0?ooo`030000003oool0oooo00l0oooo0P00000C0?ooo`800000103oool2 000000@0oooo0P0000030?ooo`@00000703oool2000000@0oooo0P0000040?ooo`800000 1@3oool3000001/0oooo0P0000040?ooo`800000103oool2000000<0oooo0`00000M0?oo o`800000103oool2000000@0oooo0P0000040?ooo`800000803oool2000000@0oooo0P00 00030?ooo`D000000@3oool00240oooo00<000000?ooo`3oool03@3oool2000001@0oooo 00@000000?ooo`3oool00000203oool010000000oooo0?ooo`0000020?ooo`030000003o ool0oooo01`0oooo00@000000?ooo`3oool00000203oool010000000oooo0?ooo`000005 0?ooo`030000003oool0oooo01T0oooo00@000000?ooo`3oool00000203oool010000000 oooo0?ooo`0000020?ooo`040000003oool0oooo000001/0oooo00@000000?ooo`3oool0 0000203oool010000000oooo0?ooo`0000020?ooo`040000003oool0oooo000001h0oooo 00@000000?ooo`3oool000002P3oool00`000000oooo0?ooo`010?ooo`008@3oool00`00 0000oooo0?ooo`0<0?ooo`030000003oool0oooo01@0oooo00@000000?ooo`3oool00000 203oool010000000oooo0?ooo`0000030?ooo`030000003oool0oooo01/0oooo00@00000 0?ooo`3oool00000203oool010000000oooo0?ooo`0000020?ooo`D000006P3oool01000 0000oooo0?ooo`0000080?ooo`040000003oool0oooo00000080oooo00@000000?ooo`3o ool000006`3oool010000000oooo0?ooo`0000080?ooo`040000003oool0oooo00000080 oooo00@000000?ooo`3oool000007P3oool010000000oooo0?ooo`00000:0?ooo`030000 003oool0oooo0040oooo000Q0?ooo`030000003oool0oooo00X0oooo0P00000G0?ooo`04 0000003oool0oooo000000P0oooo00@000000?ooo`3oool00000103oool00`000000oooo 0?ooo`0J0?ooo`040000003oool0oooo000000P0oooo00@000000?ooo`3oool000000P3o ool010000000oooo0?ooo`00000K0?ooo`040000003oool0oooo000000P0oooo00@00000 0?ooo`3oool000000P3oool3000001`0oooo00@000000?ooo`3oool00000203oool01000 0000oooo0?ooo`0000030?ooo`8000007`3oool010000000oooo0?ooo`00000:0?ooo`03 0000003oool0oooo0040oooo000Q0?ooo`030000003oool0oooo00P0oooo0P00000I0?oo o`040000003oool0oooo000000P0oooo00@000000?ooo`3oool000000P3oool010000000 oooo0?ooo`00000K0?ooo`040000003oool0oooo000000P0oooo00@000000?ooo`3oool0 00000`3oool00`000000oooo0000000K0?ooo`040000003oool0oooo000000P0oooo00@0 00000?ooo`3oool000000`3oool00`000000oooo0?ooo`0K0?ooo`040000003oool0oooo 000000P0oooo00@000000?ooo`3oool000000P3oool010000000oooo0?ooo`00000N0?oo o`040000003oool0oooo000000X0oooo00<000000?ooo`3oool00@3oool00240oooo0P00 00080?ooo`030000003oool0oooo01X0oooo0P00000:0?ooo`800000103oool2000001d0 oooo0P00000:0?ooo`8000001@3oool2000001`0oooo0P00000:0?ooo`800000103oool3 000001`0oooo0P00000:0?ooo`800000103oool200000200oooo0P00000:0?ooo`800000 0`3oool00240oooo00<000000?ooo`3oool01@3oool200000?D0oooo000Q0?ooo`030000 003oool0oooo00@0oooo00<000000?ooo`3oool0m@3oool00240oooo00<000000?ooo`3o ool00`3oool00`000000oooo0?ooo`3f0?ooo`008@3oool010000000oooo0?ooo`3oool2 00000?T0oooo000Q0?ooo`040000003oool0oooo00000?/0oooo000Q0?ooo`<00000o03o ool001/0ooooo`000003000000<0oooo000Q0?ooo`030000003oool0oooo00T0oooo00<0 00000?ooo`3oool02@3oool00`000000oooo0?ooo`090?ooo`030000003oool0oooo00X0 oooo00<000000?ooo`3oool02@3oool00`000000oooo0?ooo`090?ooo`030000003oool0 oooo00T0oooo00<000000?ooo`3oool02P3oool00`000000oooo0?ooo`090?ooo`030000 003oool0oooo00T0oooo00<000000?ooo`3oool02@3oool00`000000oooo0?ooo`0:0?oo o`030000003oool0oooo00T0oooo00<000000?ooo`3oool02@3oool00`000000oooo0?oo o`090?ooo`030000003oool0oooo00X0oooo00<000000?ooo`3oool02@3oool00`000000 oooo0?ooo`090?ooo`030000003oool0oooo00T0oooo00<000000?ooo`3oool02P3oool0 0`000000oooo0?ooo`070?ooo`008@3oool00`000000oooo0?ooo`3l0?ooo`008@3oool0 0`000000oooo0?ooo`3l0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?oo o`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3o oolQ0?ooo`00o`3ooolQ0?ooo`00\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.0135686, \ 0.473546, 0.000408522, 0.000161187}}] }, Open ]], Cell[TextData[{ "Overlay the two plots to compare them. The result is a good \ approximation for ", Cell[BoxData[ \(TraditionalForm\`a > 0.04\)]], " but is progressively more inaccurate for ", Cell[BoxData[ \(TraditionalForm\`a < 0.04\)]], ". This is because ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon] = 0.01\)]], " was used, which approximates the Dirac delta function by a finite \ width Gaussian, whose width becomes more and more noticeable when ", Cell[BoxData[ \(TraditionalForm\`a = \[ScriptCapitalO](0.01)\)]], " or less (to see this think geometrically about which region of ", Cell[BoxData[ \(TraditionalForm\`x\)]], " and ", Cell[BoxData[ \(TraditionalForm\`y\)]], " contributes to the integral when ", Cell[BoxData[ \(TraditionalForm\`a\)]], " is small). Extra computational effort (i.e. using a smaller ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\)]], ") will overcome this poor approximation problem." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Show[g1, g2, AxesLabel \[Rule] {"\<a\>", "\<Pr(a)\>"}];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 9.52381 -6.10348 13.4136 [ [.21429 .59082 -12 -9 ] [.21429 .59082 12 0 ] [.40476 .59082 -12 -9 ] [.40476 .59082 12 0 ] [.59524 .59082 -12 -9 ] [.59524 .59082 12 0 ] [.78571 .59082 -12 -9 ] [.78571 .59082 12 0 ] [.97619 .59082 -9 -9 ] [.97619 .59082 9 0 ] [1.025 .60332 0 -6.28125 ] [1.025 .60332 10 6.28125 ] [.01131 .06678 -24 -4.5 ] [.01131 .06678 0 4.5 ] [.01131 .20091 -24 -4.5 ] [.01131 .20091 0 4.5 ] [.01131 .33505 -24 -4.5 ] [.01131 .33505 0 4.5 ] [.01131 .46918 -24 -4.5 ] [.01131 .46918 0 4.5 ] [.02381 .64303 -17 0 ] [.02381 .64303 17 12.5625 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g ..25 Mabswid [ ] 0 setdash ..21429 .60332 m ..21429 .60957 L s [(0.02)] .21429 .59082 0 1 Mshowa ..40476 .60332 m ..40476 .60957 L s [(0.04)] .40476 .59082 0 1 Mshowa ..59524 .60332 m ..59524 .60957 L s [(0.06)] .59524 .59082 0 1 Mshowa ..78571 .60332 m ..78571 .60957 L s [(0.08)] .78571 .59082 0 1 Mshowa ..97619 .60332 m ..97619 .60957 L s [(0.1)] .97619 .59082 0 1 Mshowa ..125 Mabswid ..07143 .60332 m ..07143 .60707 L s ..11905 .60332 m ..11905 .60707 L s ..16667 .60332 m ..16667 .60707 L s ..2619 .60332 m ..2619 .60707 L s ..30952 .60332 m ..30952 .60707 L s ..35714 .60332 m ..35714 .60707 L s ..45238 .60332 m ..45238 .60707 L s ..5 .60332 m ..5 .60707 L s ..54762 .60332 m ..54762 .60707 L s ..64286 .60332 m ..64286 .60707 L s ..69048 .60332 m ..69048 .60707 L s ..7381 .60332 m ..7381 .60707 L s ..83333 .60332 m ..83333 .60707 L s ..88095 .60332 m ..88095 .60707 L s ..92857 .60332 m ..92857 .60707 L s ..25 Mabswid 0 .60332 m 1 .60332 L s gsave 1.025 .60332 -61 -10.2813 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20.5625 translate 1 -1 scale /g { setgray} bind def /k { setcmykcolor} bind def /p { gsave} bind def /r { setrgbcolor} bind def /w { setlinewidth} bind def /C { curveto} bind def /F { fill} bind def /L { lineto} bind def /rL { rlineto} bind def /P { grestore} bind def /s { stroke} bind def /S { show} bind def /N {currentpoint 3 -1 roll show moveto} bind def /Msf { findfont exch scalefont [1 0 0 -1 0 0 ] makefont setfont} bind \ def /m { moveto} bind def /Mr { rmoveto} bind def /Mx {currentpoint exch pop moveto} bind def /My {currentpoint pop exch moveto} bind def /X {0 rmoveto} bind def /Y {0 exch rmoveto} bind def 63.000 12.813 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 63.000 12.813 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (a) show 69.000 12.813 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 1.000 setlinewidth grestore ..02381 .06678 m ..03006 .06678 L s [(0.46)] .01131 .06678 1 0 Mshowa ..02381 .20091 m ..03006 .20091 L s [(0.47)] .01131 .20091 1 0 Mshowa ..02381 .33505 m ..03006 .33505 L s [(0.48)] .01131 .33505 1 0 Mshowa ..02381 .46918 m ..03006 .46918 L s [(0.49)] .01131 .46918 1 0 Mshowa ..125 Mabswid ..02381 .0936 m ..02756 .0936 L s ..02381 .12043 m ..02756 .12043 L s ..02381 .14726 m ..02756 .14726 L s ..02381 .17408 m ..02756 .17408 L s ..02381 .22774 m ..02756 .22774 L s ..02381 .25457 m ..02756 .25457 L s ..02381 .28139 m ..02756 .28139 L s ..02381 .30822 m ..02756 .30822 L s ..02381 .36187 m ..02756 .36187 L s ..02381 .3887 m ..02756 .3887 L s ..02381 .41553 m ..02756 .41553 L s ..02381 .44236 m ..02756 .44236 L s ..02381 .49601 m ..02756 .49601 L s ..02381 .52284 m ..02756 .52284 L s ..02381 .54966 m ..02756 .54966 L s ..02381 .57649 m ..02756 .57649 L s ..02381 .03995 m ..02756 .03995 L s ..02381 .01312 m ..02756 .01312 L s ..25 Mabswid ..02381 0 m ..02381 .61803 L s gsave ..02381 .64303 -78 -4 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20.5625 translate 1 -1 scale /g { setgray} bind def /k { setcmykcolor} bind def /p { gsave} bind def /r { setrgbcolor} bind def /w { setlinewidth} bind def /C { curveto} bind def /F { fill} bind def /L { lineto} bind def /rL { rlineto} bind def /P { grestore} bind def /s { stroke} bind def /S { show} bind def /N {currentpoint 3 -1 roll show moveto} bind def /Msf { findfont exch scalefont [1 0 0 -1 0 0 ] makefont setfont} bind \ def /m { moveto} bind def /Mr { rmoveto} bind def /Mx {currentpoint exch pop moveto} bind def /My {currentpoint pop exch moveto} bind def /X {0 rmoveto} bind def /Y {0 exch rmoveto} bind def 63.000 12.813 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 63.000 12.813 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (Pr) show %%IncludeResource: font Mathematica2Mono %%IncludeFont: Mathematica2Mono /Mathematica2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 75.000 12.813 moveto (H) show 81.000 12.813 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (a) show %%IncludeResource: font Mathematica2Mono %%IncludeFont: Mathematica2Mono /Mathematica2Mono findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 87.000 12.813 moveto (L) show 93.000 12.813 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 1.000 setlinewidth grestore 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath ..5 Mabswid ..15649 0 m ..16372 .07773 L ..17461 .16966 L ..18466 .24044 L ..20381 .34289 L ..2139 .38245 L ..22459 .41545 L ..234 .43805 L ..24413 .45672 L ..25368 .4699 L ..26243 .47883 L ..26719 .48264 L ..27227 .48597 L ..27666 .48832 L ..28141 .49036 L ..28433 .49137 L ..28701 .49216 L ..28952 .49279 L ..29228 .49335 L ..29459 .49373 L ..29707 .49405 L ..29844 .49419 L ..29969 .4943 L ..30095 .49439 L ..30213 .49445 L ..30323 .49449 L ..30442 .49452 L ..30573 .49454 L ..30694 .49454 L ..30813 .49452 L ..30943 .49449 L ..31008 .49446 L ..31079 .49443 L ..31205 .49437 L ..31452 .4942 L ..31685 .49399 L ..32122 .49348 L ..33118 .49185 L ..34198 .4895 L ..3829 .4776 L ..4223 .46445 L ..46019 .45148 L ..50053 .43763 L ..53935 .42431 L ..58063 .41018 L ..62039 .3966 L ..65863 .38356 L ..69933 .36971 L ..73851 .35641 L ..78015 .34231 L Mistroke ..82027 .32874 L ..85887 .31572 L ..89993 .3019 L ..93947 .28862 L ..97619 .2763 L Mfstroke ..02381 .60332 m ..06244 .58973 L ..10458 .57494 L ..14415 .56108 L ..18221 .54778 L ..22272 .53365 L ..26171 .52007 L ..30316 .50568 L ..34309 .49184 L ..3815 .47855 L ..42237 .46444 L ..46172 .45089 L ..49955 .43788 L ..53984 .42406 L ..57861 .41079 L ..61984 .3967 L ..65954 .38317 L ..69774 .37017 L ..73838 .35637 L ..77751 .34312 L ..81909 .32906 L ..85916 .31554 L ..89771 .30256 L ..93871 .28879 L ..97619 .27622 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg<PAVmbKF5d0`40004P0000/B000`400?l00000o`00003o o`3ooolQ0?ooo`00o`3ooolQ0?ooo`006`3oool00`000000oooo0?ooo`0N0?ooo`030000 003oool0oooo0>40oooo000K0?ooo`030000003oool0oooo01h0oooo00<000000?ooo`3o ool0h@3oool001/0oooo00<000000?ooo`3oool07P3oool00`000000oooo0?ooo`3Q0?oo o`006`3oool2000001l0oooo00<000000?ooo`3oool0h@3oool001/0oooo00<000000?oo o`3oool07P3oool00`000000oooo0?ooo`3Q0?ooo`006`3oool00`000000oooo0?ooo`0O 0?ooo`030000003oool0oooo0>00oooo000K0?ooo`030000003oool0oooo01l0oooo00<0 00000?ooo`3oool0h03oool001/0oooo00<000000?ooo`3oool07`3oool00`000000oooo 0?ooo`3P0?ooo`006`3oool00`000000oooo0?ooo`0O0?ooo`030000003oool0oooo0>00 oooo000K0?ooo`800000803oool00`000000oooo0?ooo`3P0?ooo`006`3oool00`000000 oooo0?ooo`0O0?ooo`030000003oool0oooo0>00oooo000K0?ooo`030000003oool0oooo 01l0oooo00<000000?ooo`3oool0h03oool001/0oooo00<000000?ooo`3oool07`3oool0 0`000000oooo0?ooo`3P0?ooo`006`3oool00`000000oooo0?ooo`0O0?ooo`030000003o ool0oooo0>00oooo00020?ooo`800000103oool2000000D0oooo0`000000103oool00000 0000000000050?ooo`030000003oool0oooo01l0oooo00<000000?ooo`3oool0h03oool0 00050?ooo`000000oooo0?ooo`0000002`3oool010000000oooo0?ooo`0000020?ooo`03 0000003oool0oooo0080oooo00<000000?ooo`3oool0803oool00`000000oooo0?ooo`3O 0?ooo`0000D0oooo0000003oool0oooo000000080?ooo`D0000000D0oooo0000003oool0 oooo000000040?ooo`8000008@3oool00`000000oooo0?ooo`3O0?ooo`0000D0oooo0000 003oool0oooo000000080?ooo`040000003oool0oooo00000080oooo0`0000050?ooo`03 0000003oool0oooo0200oooo00<000000?ooo`3oool0g`3oool000050?ooo`000000oooo 0?ooo`0000002@3oool00`000000oooo000000030?ooo`030000003oool0oooo00@0oooo 00<000000?ooo`3oool0803oool00`000000oooo0?ooo`3O0?ooo`000P3oool2000000/0 oooo0P0000030?ooo`<00000103oool00`000000oooo0?ooo`0P0?ooo`030000003oool0 oooo0=l0oooo000K0?ooo`030000003oool0oooo0200oooo00<000000?ooo`3oool0g`3o ool001/0oooo00<000000?ooo`3oool0803oool00`000000oooo0?ooo`3O0?ooo`006`3o ool00`000000oooo0?ooo`0P0?ooo`030000003oool0oooo0=l0oooo000K0?ooo`800000 8@3oool00`000000oooo0?ooo`3O0?ooo`006`3oool00`000000oooo0?ooo`0P0?ooo`03 0000003oool0oooo0=l0oooo000K0?ooo`030000003oool0oooo0240oooo00<000000?oo o`3oool0gP3oool001/0oooo00<000000?ooo`3oool08@3oool00`000000oooo0?ooo`3N 0?ooo`006`3oool00`000000oooo0?ooo`0Q0?ooo`030000003oool0oooo0=h0oooo000K 0?ooo`030000003oool0oooo0240oooo00<000000?ooo`3oool0gP3oool001/0oooo00<0 00000?ooo`3oool08@3oool00`000000oooo0?ooo`3N0?ooo`006`3oool200000280oooo 00<000000?ooo`3oool0gP3oool001/0oooo00<000000?ooo`3oool08@3oool00`000000 oooo0?ooo`3N0?ooo`006`3oool00`000000oooo0?ooo`0Q0?ooo`030000003oool0oooo 0=h0oooo000K0?ooo`030000003oool0oooo0240oooo00<000000?ooo`3oool0gP3oool0 01/0oooo00<000000?ooo`3oool08@3oool00`000000oooo0?ooo`3N0?ooo`006`3oool0 0`000000oooo0?ooo`0Q0?ooo`030000003oool0oooo0=h0oooo000K0?ooo`8000008P3o ool00`000000oooo0?ooo`3N0?ooo`006`3oool00`000000oooo0?ooo`0R0?ooo`030000 003oool0oooo0=d0oooo000K0?ooo`030000003oool0oooo0280oooo00<000000?ooo`3o ool0g@3oool001/0oooo00<000000?ooo`3oool08P3oool00`000000oooo0?ooo`3M0?oo o`006`3oool00`000000oooo0?ooo`0R0?ooo`030000003oool0oooo0=d0oooo000K0?oo o`030000003oool0oooo0280oooo00<000000?ooo`3oool0g@3oool001/0oooo00<00000 0?ooo`3oool08P3oool00`000000oooo0?ooo`3M0?ooo`006`3oool2000002<0oooo00<0 00000?ooo`3oool0g@3oool001/0oooo00<000000?ooo`3oool08P3oool00`000000oooo 0?ooo`3M0?ooo`006`3oool00`000000oooo0?ooo`0R0?ooo`030000003oool0oooo0=d0 oooo000K0?ooo`030000003oool0oooo02<0oooo00<000000?ooo`3oool0g03oool001/0 oooo00<000000?ooo`3oool08`3oool00`000000oooo0?ooo`3L0?ooo`000P3oool20000 00@0oooo0P0000050?ooo`<000000`3oool00`000000oooo0?ooo`030?ooo`030000003o ool0oooo02<0oooo00<000000?ooo`3oool0g03oool000050?ooo`000000oooo0?ooo`00 00002`3oool00`000000oooo0?ooo`020?ooo`030000003oool0oooo00<0oooo00<00000 0?ooo`3oool08`3oool00`000000oooo0?ooo`3L0?ooo`0000D0oooo0000003oool0oooo 000000080?ooo`D000000`3oool00`000000oooo0?ooo`030?ooo`800000903oool00`00 0000oooo0?ooo`3L0?ooo`0000D0oooo0000003oool0oooo000000080?ooo`040000003o ool0oooo000000D0oooo00<000000?ooo`3oool00P3oool00`000000oooo0?ooo`0S0?oo o`030000003oool0oooo0=`0oooo00001@3oool000000?ooo`3oool0000000T0oooo00<0 00000?ooo`0000000P3oool010000000oooo0?ooo`0000040?ooo`030000003oool0oooo 02@0oooo00<000000?ooo`3oool0f`3oool00080oooo0P00000;0?ooo`8000000P3oool4 000000@0oooo00<000000?ooo`3oool0903oool00`000000oooo0?ooo`3K0?ooo`006`3o ool00`000000oooo0?ooo`0T0?ooo`030000003oool0oooo0=/0oooo000K0?ooo`030000 003oool0oooo02@0oooo00<000000?ooo`3oool0f`3oool001/0oooo0P00000U0?ooo`03 0000003oool0oooo0=/0oooo000K0?ooo`030000003oool0oooo02@0oooo00<000000?oo o`3oool0f`3oool001/0oooo00<000000?ooo`3oool09@3oool00`000000oooo0?ooo`3J 0?ooo`006`3oool00`000000oooo0?ooo`0U0?ooo`030000003oool0oooo0=X0oooo000K 0?ooo`030000003oool0oooo02D0oooo00<000000?ooo`3oool0fP3oool001/0oooo00<0 00000?ooo`3oool09@3oool00`000000oooo0?ooo`3J0?ooo`006`3oool00`000000oooo 0?ooo`0U0?ooo`030000003oool0oooo0=X0oooo000K0?ooo`8000009`3oool00`000000 oooo0?ooo`3I0?ooo`006`3oool00`000000oooo0?ooo`0V0?ooo`030000003oool0oooo 0=T0oooo000K0?ooo`030000003oool0oooo02H0oooo00<000000?ooo`3oool0f@3oool0 01/0oooo00<000000?ooo`3oool09P3oool00`000000oooo0?ooo`3I0?ooo`006`3oool0 0`000000oooo0?ooo`0V0?ooo`030000003oool0oooo0=T0oooo000K0?ooo`030000003o ool0oooo02L0oooo00<000000?ooo`3oool0_`3oool2000001L0oooo000K0?ooo`030000 003oool0oooo02L0oooo00<000000?ooo`3oool0_@3oool2000001T0oooo000K0?ooo`80 0000:03oool00`000000oooo0?ooo`2k0?ooo`8000006`3oool001/0oooo00<000000?oo o`3oool09`3oool00`000000oooo0?ooo`2i0?ooo`8000007@3oool001/0oooo00<00000 0?ooo`3oool09`3oool00`000000oooo0?ooo`2f0?ooo`<000007`3oool001/0oooo00<0 00000?ooo`3oool0:03oool00`000000oooo0?ooo`2b0?ooo`<000008P3oool001/0oooo 00<000000?ooo`3oool0:03oool00`000000oooo0?ooo`2^0?ooo`@000009@3oool001/0 oooo00<000000?ooo`3oool0:03oool00`000000oooo0?ooo`2[0?ooo`<00000:@3oool0 01/0oooo0P00000Y0?ooo`030000003oool0oooo0:P0oooo0`00000/0?ooo`006`3oool0 0`000000oooo0?ooo`0X0?ooo`030000003oool0oooo0:@0oooo1000000_0?ooo`006`3o ool00`000000oooo0?ooo`0Y0?ooo`030000003oool0oooo0:00oooo0`00000c0?ooo`00 6`3oool00`000000oooo0?ooo`0Y0?ooo`030000003oool0oooo09h0oooo0P00000f0?oo o`006`3oool00`000000oooo0?ooo`0Y0?ooo`030000003oool0oooo09/0oooo0`00000h 0?ooo`000P3oool2000000@0oooo0P0000050?ooo`<000000P3oool2000000D0oooo00<0 00000?ooo`3oool0:@3oool00`000000oooo0?ooo`2I0?ooo`800000>`3oool000050?oo o`000000oooo0?ooo`0000002`3oool010000000oooo0?ooo`0000020?ooo`030000003o ool0oooo0080oooo00<000000?ooo`3oool0:@3oool00`000000oooo0?ooo`2F0?ooo`<0 0000?@3oool000050?ooo`000000oooo0?ooo`000000203oool5000000050?ooo`000000 oooo0?ooo`000000103oool2000002/0oooo00<000000?ooo`3oool0TP3oool300000400 oooo00001@3oool000000?ooo`3oool0000000P0oooo00@000000?ooo`3oool000000`3o ool2000000D0oooo00<000000?ooo`3oool0:P3oool00`000000oooo0?ooo`2>0?ooo`@0 0000@`3oool000050?ooo`000000oooo0?ooo`0000002@3oool00`000000oooo00000002 0?ooo`040000003oool0oooo000000@0oooo00<000000?ooo`3oool0:P3oool00`000000 oooo0?ooo`2;0?ooo`<00000A`3oool00080oooo0P00000;0?ooo`8000000`3oool20000 00D0oooo00<000000?ooo`3oool0:P3oool00`000000oooo0?ooo`280?ooo`<00000BP3o ool001/0oooo00<000000?ooo`3oool0:P3oool00`000000oooo0?ooo`240?ooo`@00000 C@3oool001/0oooo00<000000?ooo`3oool0:`3oool00`000000oooo0?ooo`200?ooo`<0 0000D@3oool001/0oooo00<000000?ooo`3oool0:`3oool00`000000oooo0?ooo`1n0?oo o`800000E03oool001/0oooo0P00000/0?ooo`030000003oool0oooo07/0oooo0`00001F 0?ooo`006`3oool00`000000oooo0?ooo`0[0?ooo`030000003oool0oooo07T0oooo0P00 001I0?ooo`006`3oool00`000000oooo0?ooo`0[0?ooo`030000003oool0oooo07H0oooo 0`00001K0?ooo`006`3oool00`000000oooo0?ooo`0/0?ooo`030000003oool0oooo0780 oooo0`00001N0?ooo`006`3oool00`000000oooo0?ooo`0/0?ooo`030000003oool0oooo 06h0oooo1000001Q0?ooo`006`3oool00`000000oooo0?ooo`0/0?ooo`030000003oool0 oooo06/0oooo0`00001U0?ooo`006`3oool00`000000oooo0?ooo`0/0?ooo`030000003o ool0oooo06P0oooo0`00001X0?ooo`006`3oool2000002h0oooo00<000000?ooo`3oool0 I03oool4000006X0oooo000K0?ooo`030000003oool0oooo02d0oooo00<000000?ooo`3o ool0H@3oool4000006d0oooo000K0?ooo`030000003oool0oooo02d0oooo00<000000?oo o`3oool0G`3oool4000006l0oooo000K0?ooo`030000003oool0oooo02h0oooo00<00000 0?ooo`3oool0F`3oool400000780oooo000K0?ooo`030000003oool0oooo02h0oooo00<0 00000?ooo`3oool0F03oool4000007D0oooo000K0?ooo`030000003oool0oooo02l0oooo 00<000000?ooo`3oool0E03oool4000007P0oooo000K0?ooo`800000<03oool00`000000 oooo0?ooo`1A0?ooo`@00000N`3oool001/0oooo00<000000?ooo`3oool0;`3oool00`00 0000oooo0?ooo`1=0?ooo`@00000O`3oool001/0oooo00<000000?ooo`3oool0<03oool0 0`000000oooo0?ooo`190?ooo`<00000P`3oool001/0oooo00<000000?ooo`3oool0<03o ool00`000000oooo0?ooo`170?ooo`800000QP3oool001/0oooo00<000000?ooo`3oool0 <03oool00`000000oooo0?ooo`140?ooo`<00000R03oool001/0oooo00<000000?ooo`3o ool0<@3oool00`000000oooo0?ooo`110?ooo`<00000RP3oool001/0oooo00<000000?oo o`3oool0<@3oool00`000000oooo0?ooo`0n0?ooo`@00000S03oool001/0oooo0P00000c 0?ooo`030000003oool0oooo03X0oooo1000002?0?ooo`006`3oool00`000000oooo0?oo o`0b0?ooo`030000003oool0oooo03L0oooo1000002B0?ooo`006`3oool00`000000oooo 0?ooo`0c0?ooo`030000003oool0oooo03<0oooo0`00002F0?ooo`006`3oool00`000000 oooo0?ooo`0d0?ooo`030000003oool0oooo02l0oooo0`00002I0?ooo`006`3oool00`00 0000oooo0?ooo`0d0?ooo`030000003oool0oooo02/0oooo1000002L0?ooo`000P3oool2 000000@0oooo0P0000050?ooo`<0000000@0oooo00000000000000001@3oool00`000000 oooo0?ooo`0e0?ooo`030000003oool0oooo02L0oooo0`00002P0?ooo`0000D0oooo0000 003oool0oooo0000000;0?ooo`030000003oool0oooo0080oooo00<000000?ooo`3oool0 0`3oool00`000000oooo0?ooo`0e0?ooo`030000003oool0oooo02D0oooo0P00002S0?oo o`0000D0oooo0000003oool0oooo000000080?ooo`D000000P3oool3000000@0oooo0P00 000g0?ooo`030000003oool0oooo0240oooo0`00002U0?ooo`0000D0oooo0000003oool0 oooo000000080?ooo`040000003oool0oooo00000080oooo00@000000?ooo`3oool00000 103oool00`000000oooo0?ooo`0g0?ooo`030000003oool0oooo01h0oooo0P00002X0?oo o`0000D0oooo0000003oool0oooo000000090?ooo`030000003oool000000080oooo00@0 00000?ooo`3oool00000103oool00`000000oooo0?ooo`0h0?ooo`030000003oool0oooo 01X0oooo0`00002Z0?ooo`000P3oool2000000/0oooo0P0000030?ooo`<00000103oool0 0`000000oooo0?ooo`0i0?ooo`030000003oool0oooo01H0oooo0`00002]0?ooo`006`3o ool00`000000oooo0?ooo`0j0?ooo`<000004@3oool400000;00oooo000K0?ooo`030000 003oool0oooo03d0oooo0`0000090?ooo`D00000]03oool001/0oooo0P0000110?ooo`/0 0000]`3oool001/0oooo00<000000?ooo`3oool0AP3oool300000;T0oooo000K0?ooo`03 0000003oool0oooo04@0oooo0P00002l0?ooo`006`3oool00`000000oooo0?ooo`110?oo o`<00000_P3oool001/0oooo00<000000?ooo`3oool0?P3oool300000<40oooo000K0?oo o`030000003oool0oooo03X0oooo100000340?ooo`006`3oool00`000000oooo0?ooo`0g 0?ooo`<00000b03oool001/0oooo0P00000f0?ooo`800000b`3oool001/0oooo00<00000 0?ooo`3oool0<P3oool300000<d0oooo000K0?ooo`030000003oool0oooo0300oooo0P00 003@0?ooo`006`3oool00`000000oooo0?ooo`0]0?ooo`<00000dP3oool001/0oooo00<0 00000?ooo`3oool0:P3oool300000=D0oooo000K0?ooo`030000003oool0oooo02H0oooo 1000003H0?ooo`006`3oool00`000000oooo0?ooo`0S0?ooo`<00000g03oool001/0oooo 0P00000R0?ooo`800000g`3oool001/0oooo00<000000?ooo`3oool07`3oool200000>40 oooo000K0?ooo`030000003oool0oooo01d0oooo0P0000030?ooo`800000103oool20000 00@0oooo0P0000030?ooo`@000006`3oool2000000@0oooo0P0000040?ooo`8000001@3o ool3000001T0oooo0P0000040?ooo`800000103oool2000000<0oooo0`00000L0?ooo`80 0000103oool2000000@0oooo0P0000040?ooo`8000007P3oool2000000@0oooo0P000003 0?ooo`D000003`3oool001/0oooo00<000000?ooo`3oool06P3oool3000000@0oooo00@0 00000?ooo`3oool00000203oool010000000oooo0?ooo`0000020?ooo`030000003oool0 oooo01/0oooo00@000000?ooo`3oool00000203oool010000000oooo0?ooo`0000050?oo o`030000003oool0oooo01L0oooo00@000000?ooo`3oool00000203oool010000000oooo 0?ooo`0000020?ooo`040000003oool0oooo000001X0oooo00@000000?ooo`3oool00000 203oool010000000oooo0?ooo`0000020?ooo`040000003oool0oooo000001`0oooo00@0 00000?ooo`3oool000002P3oool00`000000oooo0?ooo`0?0?ooo`006`3oool00`000000 oooo0?ooo`0G0?ooo`<000001`3oool010000000oooo0?ooo`0000080?ooo`040000003o ool0oooo000000<0oooo00<000000?ooo`3oool06P3oool010000000oooo0?ooo`000008 0?ooo`040000003oool0oooo00000080oooo1@00000H0?ooo`040000003oool0oooo0000 00P0oooo00@000000?ooo`3oool000000P3oool010000000oooo0?ooo`00000J0?ooo`04 0000003oool0oooo000000P0oooo00@000000?ooo`3oool000000P3oool010000000oooo 0?ooo`00000L0?ooo`040000003oool0oooo000000X0oooo00<000000?ooo`3oool03`3o ool001/0oooo00<000000?ooo`3oool04`3oool4000000X0oooo00@000000?ooo`3oool0 0000203oool010000000oooo0?ooo`0000040?ooo`030000003oool0oooo01T0oooo00@0 00000?ooo`3oool00000203oool010000000oooo0?ooo`0000020?ooo`040000003oool0 oooo000001T0oooo00@000000?ooo`3oool00000203oool010000000oooo0?ooo`000002 0?ooo`<000006`3oool010000000oooo0?ooo`0000080?ooo`040000003oool0oooo0000 00<0oooo0P00000M0?ooo`040000003oool0oooo000000X0oooo00<000000?ooo`3oool0 3`3oool001/0oooo0P00000A0?ooo`<000003P3oool010000000oooo0?ooo`0000080?oo o`040000003oool0oooo00000080oooo00@000000?ooo`3oool000006P3oool010000000 oooo0?ooo`0000080?ooo`040000003oool0oooo000000<0oooo00<000000?ooo`000000 6@3oool010000000oooo0?ooo`0000080?ooo`040000003oool0oooo000000<0oooo00<0 00000?ooo`3oool06P3oool010000000oooo0?ooo`0000080?ooo`040000003oool0oooo 00000080oooo00@000000?ooo`3oool00000703oool010000000oooo0?ooo`00000:0?oo o`030000003oool0oooo00l0oooo000K0?ooo`030000003oool0oooo00d0oooo0`00000B 0?ooo`8000002P3oool2000000@0oooo0P00000L0?ooo`8000002P3oool2000000D0oooo 0P00000J0?ooo`8000002P3oool2000000@0oooo0`00000K0?ooo`8000002P3oool20000 00@0oooo0P00000N0?ooo`8000002P3oool200000140oooo000K0?ooo`030000003oool0 oooo00/0oooo0P00003e0?ooo`006`3oool00`000000oooo0?ooo`080?ooo`<00000m`3o ool001/0oooo00<000000?ooo`3oool01@3oool300000?X0oooo000K0?ooo`030000003o ool0oooo0080oooo0`00003m0?ooo`006`3oool00`000000oooo0000000200000?P0oooo 1@0000030?ooo`005@3ooooj000000T0oooo100000040?ooo`006`3oool00`000000oooo 0?ooo`090?ooo`030000003oool0oooo00P0oooo00<000000?ooo`3oool02@3oool00`00 0000oooo0?ooo`090?ooo`030000003oool0oooo00T0oooo00<000000?ooo`3oool02@3o ool00`000000oooo0?ooo`090?ooo`030000003oool0oooo00T0oooo00<000000?ooo`3o ool02@3oool00`000000oooo0?ooo`080?ooo`030000003oool0oooo00T0oooo00<00000 0?ooo`3oool02@3oool00`000000oooo0?ooo`090?ooo`030000003oool0oooo00T0oooo 00<000000?ooo`3oool02@3oool00`000000oooo0?ooo`090?ooo`030000003oool0oooo 00T0oooo00<000000?ooo`3oool02@3oool00`000000oooo0?ooo`080?ooo`030000003o ool0oooo00T0oooo00<000000?ooo`3oool0403oool00`000000oooo0?ooo`020?ooo`00 6`3oool00`000000oooo0?ooo`0/0?ooo`030000003oool0oooo02d0oooo00<000000?oo o`3oool0;03oool00`000000oooo0?ooo`0]0?ooo`030000003oool0oooo02`0oooo00<0 00000?ooo`3oool03P3oool3000000@0oooo000K0?ooo`030000003oool0oooo0?l0oooo 0`3oool001/0oooo00<000000?ooo`3oool0o`3oool30?ooo`00o`3ooolQ0?ooo`00o`3o oolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?oo o`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00703oool00`000000oooo0?ooo`060?ooo`03 0000003oool0oooo0?P0oooo000K0?ooo`030000003oool0oooo00P0oooo00<000000?oo o`3oool0m`3oool000`0oooo0`0000040?ooo`@000000`3oool00`000000oooo0?ooo`02 0?ooo`D000000`3oool00`000000oooo0?ooo`3f0?ooo`003@3oool00`000000oooo0?oo o`040?ooo`030000003oool0oooo00<0oooo00<000000?ooo`3oool00P3oool4000000@0 oooo00<000000?ooo`3oool0mP3oool000d0oooo0`0000040?ooo`030000003oool0oooo 00<0oooo00<000000?ooo`3oool01@3oool00`000000oooo0?ooo`020?ooo`030000003o ool0oooo0?H0oooo000=0?ooo`040000003oool0oooo00000080oooo100000030?ooo`03 0000003oool0oooo00<0oooo0`0000040?ooo`030000003oool0oooo0?H0oooo000=0?oo o`040000003oool0oooo000000X0oooo00<000000?ooo`3oool0203oool00`000000oooo 0?ooo`3g0?ooo`00303oool4000000`0oooo00<000000?ooo`3oool01P3oool00`000000 oooo0?ooo`3h0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00o`3o oolQ0?ooo`00\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.0114099, \ 0.454441, 0.000420706, 0.000298706}}] }, Open ]], Cell[TextData[{ "Variants of the above can be used to map other choices of PDF ", Cell[BoxData[ \(TraditionalForm\`Pr(x, y)\)]], " using other choices of mapping function ", Cell[BoxData[ \(TraditionalForm\`f(x, y)\)]], "." }], "Text"] }, Open ]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 941}}, WindowSize->{621, 740}, WindowMargins->{{211, Automatic}, {77, Automatic}} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1776, 53, 46, 0, 95, "Title"], Cell[1825, 55, 144, 6, 167, "Subtitle"], Cell[1972, 63, 1312, 45, 144, "Text"], Cell[3287, 110, 129, 5, 33, "Text"], Cell[3419, 117, 168, 3, 55, "Input"], Cell[3590, 122, 54, 0, 33, "Text"], Cell[CellGroupData[{ Cell[3669, 126, 136, 3, 30, "Input"], Cell[3808, 131, 53, 1, 29, "Output"] }, Open ]], Cell[3876, 135, 328, 9, 52, "Text"], Cell[4207, 146, 62, 1, 30, "Input"], Cell[4272, 149, 320, 9, 52, "Text"], Cell[4595, 160, 158, 3, 51, "Input"], Cell[4756, 165, 217, 4, 52, "Text"], Cell[4976, 171, 62, 1, 30, "Input"], Cell[5041, 174, 651, 21, 71, "Text"], Cell[CellGroupData[{ Cell[5717, 199, 257, 7, 50, "Input"], Cell[5977, 208, 17353, 467, 186, 4002, 280, "GraphicsData", \ "PostScript", "Graphics"] }, Open ]], Cell[23345, 678, 462, 16, 52, "Text"], Cell[CellGroupData[{ Cell[23832, 698, 237, 6, 50, "Input"], Cell[24072, 706, 18878, 462, 186, 3572, 248, "GraphicsData", \ "PostScript", "Graphics"] }, Open ]], Cell[42965, 1171, 1730, 47, 244, "Text"], Cell[44698, 1220, 232, 8, 33, "Text"], Cell[CellGroupData[{ Cell[44955, 1232, 195, 5, 43, "Input"], Cell[45153, 1239, 16839, 424, 186, 3350, 235, "GraphicsData", \ "PostScript", "Graphics"] }, Open ]], Cell[62007, 1666, 1016, 28, 128, "Text"], Cell[CellGroupData[{ Cell[63048, 1698, 101, 2, 30, "Input"], Cell[63152, 1702, 23135, 628, 186, 7472, 409, "GraphicsData", \ "PostScript", "Graphics"] }, Open ]], Cell[86302, 2333, 257, 8, 52, "Text"] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)