(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. 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@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[ 364992, 11899]*) (*NotebookOutlinePosition[ 383351, 12560]*) (* CellTagsIndexPosition[ 383307, 12556]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ "Mathematical Statistics with ", StyleBox["Mathematica", FontSlant->"Italic"] }], "Title", TextAlignment->Center, CounterAssignments->{{"Chapter", 1}}], Cell["Colin Rose and Murray D. Smith", "Subtitle", TextAlignment->Center], Cell[TextData[{ StyleBox["Abstract", FontWeight->"Bold"], "\nWe present a unified approach for doing mathematical statistics with ", StyleBox["Mathematica", FontSlant->"Italic"], ". At one extreme, our package ", StyleBox["PDF", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " empowers even the \"statistically challenged\" with the ability to \ perform complicated operations without realizing it. At the other extreme, it \ enables the professional statistician to tackle tricky multivariate \ distributions, generating functions, transformations, symbolic maximum \ likelihood estimation, unbiased estimation, checking (and correcting) of \ textbook formulae, and so on. By taking full advantage of the latest v4 \ Assumptions technology, the ", StyleBox["PDF", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " package can produce exceptionally clean and neat symbolic output. " }], "Text", CellMargins->{{53, 56.1875}, {Inherited, Inherited}}, FontSize->10], Cell[TextData[{ StyleBox["Dr Colin Rose \t ", FontSize->9], StyleBox["Theoretical Research Institute", FontSize->9, FontSlant->"Italic"], StyleBox[", Sydney\tcolin@tri.org.au\nDr Murray D. Smith\t ", FontSize->9], StyleBox["University of Sydney", FontSize->9, FontSlant->"Italic"], StyleBox[" \t\t\t \tMurray.Smith@econ.usyd.edu.au", FontSize->9] }], "Text", CellMargins->{{53, 56.1875}, {80, Inherited}}, TextAlignment->Left, TextJustification->0, FontSize->11], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Chapter"], "\tIntroduction" }], "Section"], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Chapter"], ".", CounterBox["Section"], "\tA New Approach" }], "Subsubsection"], Cell[TextData[{ "The use of computer software in statistics is far from knew. Indeed, \ literally hundreds of statistical computer programs exist. Yet, underlying \ existing programs is almost always a numerical / graphical view of the world. \ ", StyleBox["Mathematica", FontSlant->"Italic"], " can easily handle the numerical and graphical sides, but it offers in \ addition an extremely powerful and flexible symbolic computer algebra system. \ The ", StyleBox["PDF", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " software package builds upon that symbolic engine to create a \ sophisticated toolset specially designed for doing mathematical statistics. \ This forms the basis of a Springer-Verlag text entitled ", StyleBox["Mathematical Statistics with Mathematica", FontSlant->"Italic"], " (500 pages) that ships with CD-ROM, custom palettes, online help, and \ live interactive chapters. This provides a simple and unified approach for \ doing mathematical statistics with ", StyleBox["Mathematica", FontSlant->"Italic"], ", suited to those in statistics, econometrics, engineering, physics, \ psychometrics, economics, finance, biometrics - indeed across the full ambit \ of the physical/social sciences and across both the pure and the applied \ domains. It is expected to be available by January 2000. " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Chapter"], ".2\tFeatures" }], "Subsubsection"], Cell[TextData[{ StyleBox["Features include", FontWeight->"Bold"], ": * a complete suite of functions for manipulating probability density \ functions, * symbolic maximum likelihood estimation, * numerical maximum \ likelihood estimation, * automated symbolic Pearson curve fitting, * Johnson \ fitting, * Gram-Charlier expansions, * nonparametric kernel density \ estimation, * moment conversion formulae (convert automatically between \ cumulants, raw moments, central moments, and factorial moments: univariate \ and multivariate), * random number generation for any discrete distribution, \ * fully automated transformations (functions of random variables), * \ asymptotics, * decision theory (order statistics, mean square error, ...), * \ unbiased estimation (h-statistics, k-statistics, polykays). The ", StyleBox["PDF", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " Application Package can replace many reference works and extends the \ analysis to problems of arbitrary high order. " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Chapter"], ".3\tDesign Philosophy" }], "Subsubsection"], Cell[TextData[{ "The ", StyleBox["PDF", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " package has been designed with two central goals: it sets out to be ", StyleBox["general", FontWeight->"Bold"], StyleBox[", ", FontSlant->"Italic"], "and it strives to be", StyleBox[" 'delightfully", FontWeight->"Bold"], " ", StyleBox["simple'", FontWeight->"Bold"], StyleBox[".", FontSlant->"Italic"] }], "Text"], Cell[TextData[{ "By ", StyleBox["general", FontWeight->"Bold"], ", we mean that it should ", StyleBox["not", FontSlant->"Italic"], " be limited to a set of special or well-known textbook distributions. It \ should ", StyleBox["not", FontSlant->"Italic"], " operate like a textbook appendix with prepared 'crib sheet' answers. \ Rather, it should know how to solve problems from first principles. It should \ seamlessly handle: univariate ", StyleBox["and", FontSlant->"Italic"], " multivariate distributions, continuous ", StyleBox["and", FontSlant->"Italic"], " discrete random variables, distributions of functions of random \ variables, all with ", StyleBox["and", FontSlant->"Italic"], " without parameters \[Dash] and distributions no-one has ever thought of \ before. " }], "IndentText"], ver thought of \ before. " }], "Indendelightfully simple", FontWeight->"Bold"], ", we mean both (i) easy to use, and (ii) able to solve problems that seem \ difficult but which are formally quite simple. Consider, for instance, \ playing a devilish game of chess against a strong chess computer: in the \ middle of the game, after a short pause, the computer blandly announces \ \"Mate in 16 moves\". The problem it has solved might seem fantastically \ difficult, but it is really just a 'delightfully simple' finite problem that \ is conceptually no different to looking just 2 moves ahead. Of course, as \ soon as one has a tool for solving such problems, the notion of what is \ 'difficult' changes completely. A pocket calculator is certainly a \ delightfully simple device: it is easy to use, and it can solve tricky \ problems that were previously thought to be difficult. But today, few people \ bother to ponder at the marvel of a calculator any more, and we now generally \ spend our time either using such tools, or trying to solve higher-order \ conceptual problems ... and so, we are certain, it will be with mathematical \ statistics too. In fact, while much of the material traditionally studied in \ mathematical statistics courses may appear to many to be difficult, such \ material is often really just delightfully simple. Normally, all we want is \ an expectation, or a probability, or a transformation. But once we are armed \ with a computerised expectation operator, we can find any kind of expectation \ including the mean, variance, skewness, kurtosis, moment generating function, \ characteristic function, raw moments, central moments, cumulants, probability \ generating function, factorial moment generating function, and so on. \ Normally, many of these calculations are not attempted in undergraduate \ texts, because the mechanics are deemed too hard. Any yet, underlying all of \ them is just the delightfully simple expectation operator." }], "IndentText"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2\tCore Functions", "Section"], Cell[CellGroupData[{ Cell["2.1\tGetting started", "Subsubsection"], Cell[TextData[{ "The ", StyleBox["PDF", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " package adds over a hundred new functions to ", StyleBox["Mathematica", FontSlant->"Italic"], ". But 95% of the time, we can get by with just 4 of them: " }], "Text"], Cell[BoxData[ FormBox[ RowBox[{" ", StyleBox[ FrameBox[ RowBox[{" ", StyleBox[GridBox[{ {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"abbreviation\\\\\\\"\\\ \", \\\"SO10\\\"]\\) \"\>", \ "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"description\\\\\\\"\\\", \\\"SO10\\\"]\\) \ \"\>"}, { RowBox[{ RowBox[{ StyleBox["PlotDensity", FontFamily->"Courier"], StyleBox["[", FontFamily->"Courier"], StyleBox[" ", FontSlant->"Italic"], StyleBox["f", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier"]}], " "}], \(Plotting\ \((automated)\)\)}, { RowBox[{ RowBox[{ StyleBox["Expect", FontFamily->"Courier"], StyleBox["[", FontFamily->"Courier"], RowBox[{ StyleBox["x", FontSlant->"Italic"], ",", StyleBox["f", FontSlant->"Italic"]}], StyleBox[" ", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier"]}], " "}], RowBox[{"Expectation", " ", "operator", FormBox[\(\(\ \ \ \ \ \ \ \ \)\(E[X]\)\), "TraditionalForm"], " "}]}, { RowBox[{ StyleBox["Prob", FontFamily->"Courier"], StyleBox["[", FontFamily->"Courier"], RowBox[{ StyleBox["x", FontSlant->"Italic"], ",", StyleBox["f", FontSlant->"Italic"]}], StyleBox["]", FontFamily-> "Courier"]}], \(Probability\ \ \ \ \ \ \ \ \t\ \ \ \ \ \ \(P(X \[LessEqual] x)\)\)}, { RowBox[{ RowBox[{ StyleBox["Transform", FontFamily->"Courier"], StyleBox["[", FontFamily->"Courier"], RowBox[{ StyleBox["eqn", FontSlant->"Italic"], ",", StyleBox["f", FontSlant->"Italic"]}], StyleBox[" ", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier"]}], StyleBox[" ", FontFamily->"Courier"]}], "Transformations"} }, ColumnAlignments->{Left}], GridBoxOptions->{RowLines->{True, False}}], StyleBox[" ", GridBoxOptions->{RowLines->{True, False}}]}]], "4ColumnBox", Background->RGBColor[1, 1, 0.900008]], StyleBox[" ", "4ColumnBox", Background->RGBColor[1, 1, 0.900008]]}], TraditionalForm]], "Text", ShowStringCharacters->False], Cell[TextData[{ StyleBox["Table ", FontWeight->"Bold"], StyleBox[ CounterBox["Table"], FontWeight->"Bold"], StyleBox[": ", FontWeight->"Bold"], "Core functions for a random variable ", Cell[BoxData[ \(TraditionalForm\`X\)]], " with density ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]] }], "TableLabel"], Cell[TextData[{ "This ability to handle plotting, expectations, probability, and \ transformations, with just 4 functions, makes the ", StyleBox["PDF", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " system very easy to use, even for those not familiar with ", StyleBox["Mathematica", FontSlant->"Italic"], ". " }], "Text"], Cell[TextData[{ "To illustrate, let us suppose the continuous random variable ", Cell[BoxData[ \(TraditionalForm\`X\)]], " has probability density function (pdf) ", Cell[BoxData[ \(TraditionalForm\`f(x)\ = \ \ 1\/\(\[Pi] \(\@\( 1 - x\)\) \@x\)\)]], ", where ", Cell[BoxData[ \(TraditionalForm\`x\ \[Element] \ \((0, 1)\)\)]], ". In ", StyleBox["Mathematica", FontSlant->"Italic"], ", we enter this as: " }], "Text"], Cell[BoxData[ \(f\ = 1\/\(\[Pi]\ \@\(1 - x\)\ \@x\); \ \ \ \ \ \ \ \ \ \ \ domain[ f]\ = \ {x, \ 0, \ 1};\)], "Input"], Cell[TextData[{ "This is known as the Arc-Sine distribution. Here is a plot of ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(PlotDensity[f];\)\)], "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 0.0238094 0.952381 0 0.0874933 [ [.21429 -0.0125 -8.25 -12 ] [.21429 -0.0125 8.25 0 ] [.40476 -0.0125 -8.25 -12 ] [.40476 -0.0125 8.25 0 ] [.59524 -0.0125 -8.25 -12 ] [.59524 -0.0125 8.25 0 ] [.78571 -0.0125 -8.25 -12 ] [.78571 -0.0125 8.25 0 ] [.97619 -0.0125 -4.5 -12 ] [.97619 -0.0125 4.5 0 ] [1.025 0 0 -6 ] [1.025 0 8.6875 6 ] [.01131 .08749 -9 -6 ] [.01131 .08749 0 6 ] [.01131 .17499 -9 -6 ] [.01131 .17499 0 6 ] [.01131 .26248 -9 -6 ] [.01131 .26248 0 6 ] [.01131 .34997 -9 -6 ] [.01131 .34997 0 6 ] [.01131 .43747 -9 -6 ] [.01131 .43747 0 6 ] [.01131 .52496 -9 -6 ] [.01131 .52496 0 6 ] [.01131 .61245 -9 -6 ] [.01131 .61245 0 6 ] [.02381 .64303 -4.8125 0 ] [.02381 .64303 4.8125 12.0625 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 0 m .21429 .00625 L s gsave .21429 -0.0125 -69.25 -16 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (0.2) show 1.000 setlinewidth grestore .40476 0 m .40476 .00625 L s gsave .40476 -0.0125 -69.25 -16 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (0.4) show 1.000 setlinewidth grestore .59524 0 m .59524 .00625 L s gsave .59524 -0.0125 -69.25 -16 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (0.6) show 1.000 setlinewidth grestore .78571 0 m .78571 .00625 L s gsave .78571 -0.0125 -69.25 -16 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (0.8) show 1.000 setlinewidth grestore .97619 0 m .97619 .00625 L s gsave .97619 -0.0125 -65.5 -16 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (1) show 1.000 setlinewidth grestore .125 Mabswid .07143 0 m .07143 .00375 L s .11905 0 m .11905 .00375 L s .16667 0 m .16667 .00375 L s .2619 0 m .2619 .00375 L s .30952 0 m .30952 .00375 L s .35714 0 m .35714 .00375 L s .45238 0 m .45238 .00375 L s .5 0 m .5 .00375 L s .54762 0 m .54762 .00375 L s .64286 0 m .64286 .00375 L s .69048 0 m .69048 .00375 L s .7381 0 m .7381 .00375 L s .83333 0 m .83333 .00375 L s .88095 0 m .88095 .00375 L s .92857 0 m .92857 .00375 L s .25 Mabswid 0 0 m 1 0 L s gsave 1.025 0 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 63.250 13.000 moveto %%IncludeResource: font Times-Italic %%IncludeFont: Times-Italic /Times-Italic findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (x) show 67.688 13.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman 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 .08749 m .03006 .08749 L s gsave .01131 .08749 -70 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (1) show 1.000 setlinewidth grestore .02381 .17499 m .03006 .17499 L s gsave .01131 .17499 -70 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (2) show 1.000 setlinewidth grestore .02381 .26248 m .03006 .26248 L s gsave .01131 .26248 -70 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (3) show 1.000 setlinewidth grestore .02381 .34997 m .03006 .34997 L s gsave .01131 .34997 -70 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (4) show 1.000 setlinewidth grestore .02381 .43747 m .03006 .43747 L s gsave .01131 .43747 -70 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (5) show 1.000 setlinewidth grestore .02381 .52496 m .03006 .52496 L s gsave .01131 .52496 -70 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (6) show 1.000 setlinewidth grestore .02381 .61245 m .03006 .61245 L s gsave .01131 .61245 -70 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (7) show 1.000 setlinewidth grestore .125 Mabswid .02381 .0175 m .02756 .0175 L s .02381 .035 m .02756 .035 L s .02381 .0525 m .02756 .0525 L s .02381 .06999 m .02756 .06999 L s .02381 .10499 m .02756 .10499 L s .02381 .12249 m .02756 .12249 L s .02381 .13999 m .02756 .13999 L s .02381 .15749 m .02756 .15749 L s .02381 .19249 m .02756 .19249 L s .02381 .20998 m .02756 .20998 L s .02381 .22748 m .02756 .22748 L s .02381 .24498 m .02756 .24498 L s .02381 .27998 m .02756 .27998 L s .02381 .29748 m .02756 .29748 L s .02381 .31498 m .02756 .31498 L s .02381 .33247 m .02756 .33247 L s .02381 .36747 m .02756 .36747 L s .02381 .38497 m .02756 .38497 L s .02381 .40247 m .02756 .40247 L s .02381 .41997 m .02756 .41997 L s .02381 .45497 m .02756 .45497 L s .02381 .47246 m .02756 .47246 L s .02381 .48996 m .02756 .48996 L s .02381 .50746 m .02756 .50746 L s .02381 .54246 m .02756 .54246 L s .02381 .55996 m .02756 .55996 L s .02381 .57746 m .02756 .57746 L s .02381 .59495 m .02756 .59495 L s .25 Mabswid .02381 0 m .02381 .61803 L s gsave .02381 .64303 -65.8125 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20.0625 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 64.438 13.000 moveto %%IncludeResource: font Times-Italic %%IncludeFont: Times-Italic /Times-Italic findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (f) show 68.625 13.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman 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 .02584 .61803 m .02605 .57425 L .02729 .46157 L .02846 .39967 L .02954 .36024 L .03053 .33265 L .03279 .28823 L .03396 .27124 L .03506 .2578 L .03756 .23348 L .04 .21547 L .04262 .20017 L .04793 .17727 L .05297 .16165 L .0575 .15076 L .06244 .14117 L .07302 .12581 L .08429 .1142 L .09493 .10594 L .10458 .09997 L .12383 .09084 L .14465 .08367 L .16458 .07847 L .18565 .07415 L .22362 .0684 L .24313 .06615 L .26405 .06413 L .30541 .06103 L .34525 .0589 L .38358 .05744 L .40336 .05688 L .42436 .05642 L .43373 .05625 L .44383 .05609 L .4533 .05597 L .46211 .05588 L .47169 .0558 L .47675 .05577 L .48215 .05574 L .48488 .05573 L .48784 .05572 L .49063 .05571 L .49196 .05571 L .49321 .05571 L .49442 .0557 L .49553 .0557 L .49676 .0557 L .49744 .0557 L .49809 .0557 L .49937 .0557 L Mistroke .50056 .0557 L .50186 .0557 L .50259 .0557 L .50325 .0557 L .50446 .0557 L .50559 .0557 L .50814 .05571 L .51064 .05571 L .5133 .05572 L .51814 .05574 L .52254 .05576 L .53324 .05584 L .54337 .05593 L .56332 .0562 L .58175 .05654 L .60046 .05698 L .62107 .05759 L .66133 .0592 L .70007 .06138 L .74126 .06461 L .76183 .06668 L .78094 .06898 L .8191 .07504 L .83959 .07946 L .85819 .08453 L .87742 .09135 L .88733 .09575 L .89823 .10159 L .90784 .1079 L .91814 .11641 L .92903 .12837 L .93431 .13583 L .9392 .14414 L .94352 .15301 L .94829 .16515 L .95309 .18102 L .95575 .19216 L .95819 .20451 L .96032 .21756 L .96265 .23527 L .965 .25843 L .96629 .27461 L .96747 .29238 L .96852 .31151 L .96965 .33728 L .97085 .37302 L .97199 .42005 L .97305 .48558 L .97417 .60564 L Mfstroke .97417 .60564 m .97422 .61803 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{277.625, 170.938}, ImageMargins->{{52.875, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggoo000DOol00`00Oomoo`05Ool00`00Oomoo`0BOol2002WOol2000iOol001Aoo`03001oogoo 00Eoo`03001oogoo011oo`800:]oo`<003Ioo`004goo0P001goo00<007ooOol03Woo0P00/7oo0P00 =7oo000DOol00`00Oomoo`05Ool2000=Ool2002dOol00`00Oomoo`0aOol001aoo`03001oogoo00]o o`03001oogoo0;Eoo`03001oogoo031oo`0077oo00<007ooOol02Woo00<007ooOol0]goo00<007oo Ool0;goo000LOol00`00Oomoo`09Ool00`00Oomoo`2hOol00`00Oomoo`0_Ool001aoo`8000Uoo`03 001oogoo0;Yoo`03001oogoo02ioo`0077oo00<007ooOol027oo00<007ooOol0^goo00<007ooOol0 ;Goo000LOol00`00Oomoo`07Ool00`00Oomoo`2mOol00`00Oomoo`0/Ool001aoo`8000Qoo`03001o ogoo0;ioo`03001oogoo02]oo`0077oo00<007ooOol01Woo00<007ooOol0`7oo00<007ooOol0:Woo 000LOol00`00Oomoo`05Ool00`00Oomoo`31Ool00`00Oomoo`0ZOol001aoo`03001oogoo00Aoo`03 001oogoo0<=oo`03001oogoo02Uoo`0077oo0P001Goo00<007ooOol0`goo00<007ooOol0:Goo000L Ool00`00Oomoo`03Ool00`00Oomoo`35Ool00`00Oomoo`0XOol0019oo`@000Ioo`03001oogoo00=o o`03001oogoo0Ool00`00Oomoo`0UOol001=oo`8000Moo`03001oo`000"], ImageRangeCache->{{{147, 423.625}, {677.188, 507.25}} -> {-0.827926, \ 25.2941, 0.00471851, 0.0513619}}] }, Open ]], Cell[TextData[{ StyleBox["Fig. ", FontWeight->"Bold"], StyleBox[ CounterBox["Figure"], FontWeight->"Bold"], StyleBox[": ", FontWeight->"Bold"], " The Arc-Sine pdf" }], "FigureLabel"], Cell[TextData[{ "Here is the cumulative distribution function (cdf), ", Cell[BoxData[ \(TraditionalForm\`P(X \[LessEqual] x)\)]], ", which also provides the clue to the naming of this distribution:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Prob[x, \ f]\)], "Input"], Cell[BoxData[ \(\(2\ ArcSin[\@x]\)\/\[Pi]\)], "Output"] }, Open ]], Cell[TextData[{ "The mean, ", Cell[BoxData[ \(TraditionalForm\`E[X]\)]], ", is:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Expect[x, \ f]\)], "Input"], Cell[BoxData[ \(1\/2\)], "Output"] }, Open ]], Cell[TextData[{ "while the variance of ", Cell[BoxData[ \(TraditionalForm\`X\)]], " is:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Var[x, \ f]\)], "Input"], Cell[BoxData[ \(1\/8\)], "Output"] }, Open ]], Cell[TextData[{ "The ", Cell[BoxData[ \(TraditionalForm\`r\^th\)]], " moment of ", Cell[BoxData[ \(TraditionalForm\`X\)]], " is ", Cell[BoxData[ \(TraditionalForm\`E[X\^r]\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Expect[x\^r, \ f]\)], "Input"], Cell[BoxData[ \(\[CapitalGamma][1\/2 + r]\/\(\@\[Pi]\ \[CapitalGamma][1 + r]\)\)], \ "Output"] }, Open ]], Cell[TextData[{ "The moment generating function (mgf) of ", Cell[BoxData[ \(TraditionalForm\`X\)]], " is ", Cell[BoxData[ \(TraditionalForm\`E\ [\[ExponentialE]\^\(t\ X\)]\)]], " :" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Expect[\[ExponentialE]\^\(t\ x\), \ f]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(t/2\)\ BesselI[0, t\/2]\)], "Output"] }, Open ]], Cell[TextData[{ "Now consider the transformation to a new random variable ", Cell[BoxData[ \(TraditionalForm\`Y\)]], " such that ", Cell[BoxData[ \(TraditionalForm\`Y = \@X\)]], ". By using the ", StyleBox["Transform", FontFamily->"Courier"], " and ", StyleBox["TransformExtremum", FontFamily->"Courier"], " functions, the pdf of ", Cell[BoxData[ \(TraditionalForm\`Y\)]], ", say ", Cell[BoxData[ \(TraditionalForm\`g(y)\)]], ", and the domain of its support can be found: " }], "IndentText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ g\ \ = \ Transform[y\ \[Equal] \@x, \ f]\), "\[IndentingNewLine]", \(domain[g]\ = \ TransformExtremum[y\ \[Equal] \@x, \ f]\)}], "Input"], Cell[BoxData[ \(\(2\ y\)\/\(\[Pi]\ \@\(y\^2 - y\^4\)\)\)], "Output"], /\(\[Pi]\ \@\(y\^2 - y\^4\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "So, we have started out with a quite arbitrary pdf ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], ", transformed it to a new one ", Cell[BoxData[ \(TraditionalForm\`g(y)\)]], ", and since both density ", StyleBox["g", FontSlant->"Italic"], " and its ", StyleBox["domain", FontFamily->"Courier"], " have been inputted into ", StyleBox["Mathematica", FontSlant->"Italic"], ", we can also apply the ", StyleBox["PDF", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " tool set to density ", Cell[BoxData[ \(TraditionalForm\`g\)]], ". " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "2.2\tWorking with parameters ", StyleBox["( Assumptions technology )", FontSize->10, FontWeight->"Plain"] }], "Subsubsection"], Cell[TextData[{ "The ", StyleBox["PDF", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " package has been designed to seamlessly support parameters. It does so by \ taking full advantage of the new Assumptions technology introduced in Version \ 4 of ", StyleBox["Mathematica", FontSlant->"Italic"], ". To illustrate, let us consider the familiar Normal distribution with \ mean \[Mu] and variance ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], ". That is, let ", Cell[BoxData[ \(TraditionalForm\`X~\(N(\[Mu], \ \[Sigma]\^2)\)\)]], ", where ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\ \[Element] \ \[DoubleStruckCapitalR]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Sigma] > 0\)]], ". We enter the pdf ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], " in the standard way, but this time we have some extra information about \ the parameters \[Mu] and \[Sigma] which can be added to the end of the ", StyleBox["domain[f]", FontFamily->"Courier"], " statement:" }], "Text"], Cell[BoxData[{ \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(f\ \ \ = \ \ \ \(1\/\(\[Sigma] \ \@\( 2\ \[Pi]\)\)\) Exp[\(-\(\((x\ - \ \[Mu])\)\^2\/\(2\ \[Sigma]\^2\)\)\)];\)\ \ \ \ \ \ \ \), "\[IndentingNewLine]", \(\(domain[ f]\ = \ {x, \ \(-\[Infinity]\), \ \[Infinity]}\ && \ {\[Mu] \ \[Element] Reals, \ \[Sigma] > 0};\)\)}], "Input", GeneratedCell->False], Cell[TextData[{ "From now on, the assumptions about \[Mu] and \[Sigma] will be 'attached' \ to density ", Cell[BoxData[ \(TraditionalForm\`f\)]], ", so that whenever we operate on density ", Cell[BoxData[ \(TraditionalForm\`f\)]], " with a ", StyleBox["PDF", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " package function, these assumptions will be automatically applied in the \ background. With this new technology, the ", StyleBox["PDF", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " package can usually produce remarkably crisp textbook-style answers, even \ when working with very complicated distributions." }], "Text"], Cell[TextData[{ "The ", StyleBox["PDF", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " package function, ", StyleBox["PlotDensity", FontFamily->"Courier"], ", makes it extremely easy to examine the effect of changing parameter \ values. For instance, here is the pdf plotted at three different values of \ \[Sigma]: " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(PlotDensity[ f\ /. \ {\[Mu] \[Rule] 0, \ \[Sigma] \[Rule] \ {1, \ 2, \ 3}}];\)\)], "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 0.499998 0.0682258 0 1.5114 [ [.09064 -0.0125 -7.34375 -11 ] [.09064 -0.0125 7.34375 0 ] [.2271 -0.0125 -7.34375 -11 ] [.2271 -0.0125 7.34375 0 ] [.36355 -0.0125 -7.34375 -11 ] [.36355 -0.0125 7.34375 0 ] [.63645 -0.0125 -4.25 -11 ] [.63645 -0.0125 4.25 0 ] [.7729 -0.0125 -4.25 -11 ] [.7729 -0.0125 4.25 0 ] [.90935 -0.0125 -4.25 -11 ] [.90935 -0.0125 4.25 0 ] [1.025 0 0 -5.5 ] [1.025 0 8.25 5.5 ] [.4875 .15114 -15.25 -5.5 ] [.4875 .15114 0 5.5 ] [.4875 .30228 -15.25 -5.5 ] [.4875 .30228 0 5.5 ] [.4875 .45342 -15.25 -5.5 ] [.4875 .45342 0 5.5 ] [.4875 .60456 -15.25 -5.5 ] [.4875 .60456 0 5.5 ] [.5 .64303 -4.53125 0 ] [.5 .64303 4.53125 11 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .09064 0 m .09064 .00625 L s gsave .09064 -0.0125 -68.3438 -15 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Math1 %%IncludeFont: Math1 /Math1 findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (-) show 69.188 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (6) show 1.000 setlinewidth grestore .2271 0 m .2271 .00625 L s gsave .2271 -0.0125 -68.3438 -15 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Math1 %%IncludeFont: Math1 /Math1 findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (-) show 69.188 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (4) show 1.000 setlinewidth grestore .36355 0 m .36355 .00625 L s gsave .36355 -0.0125 -68.3438 -15 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Math1 %%IncludeFont: Math1 /Math1 findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (-) show 69.188 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (2) show 1.000 setlinewidth grestore .63645 0 m .63645 .00625 L s gsave .63645 -0.0125 -65.25 -15 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (2) show 1.000 setlinewidth grestore .7729 0 m .7729 .00625 L s gsave .7729 -0.0125 -65.25 -15 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (4) show 1.000 setlinewidth grestore .90935 0 m .90935 .00625 L s gsave .90935 -0.0125 -65.25 -15 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (6) show 1.000 setlinewidth grestore .125 Mabswid .12476 0 m .12476 .00375 L s .15887 0 m .15887 .00375 L s .19298 0 m .19298 .00375 L s .26121 0 m .26121 .00375 L s .29532 0 m .29532 .00375 L s .32943 0 m .32943 .00375 L s .39766 0 m .39766 .00375 L s .43177 0 m .43177 .00375 L s .46589 0 m .46589 .00375 L s .53411 0 m .53411 .00375 L s .56822 0 m .56822 .00375 L s .60234 0 m .60234 .00375 L s .67056 0 m .67056 .00375 L s .70468 0 m .70468 .00375 L s .73879 0 m .73879 .00375 L s .80701 0 m .80701 .00375 L s .84113 0 m .84113 .00375 L s .87524 0 m .87524 .00375 L s .05653 0 m .05653 .00375 L s .02242 0 m .02242 .00375 L s .94347 0 m .94347 .00375 L s .97758 0 m .97758 .00375 L s .25 Mabswid 0 0 m 1 0 L s gsave 1.025 0 -61 -9.5 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 63.250 12.000 moveto %%IncludeResource: font Times-Italic %%IncludeFont: Times-Italic /Times-Italic findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (x) show 67.250 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 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 .5 .15114 m .50625 .15114 L s gsave .4875 .15114 -76.25 -9.5 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (0.1) show 1.000 setlinewidth grestore .5 .30228 m .50625 .30228 L s gsave .4875 .30228 -76.25 -9.5 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (0.2) show 1.000 setlinewidth grestore .5 .45342 m .50625 .45342 L s gsave .4875 .45342 -76.25 -9.5 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (0.3) show 1.000 setlinewidth grestore .5 .60456 m .50625 .60456 L s gsave .4875 .60456 -76.25 -9.5 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (0.4) show 1.000 setlinewidth grestore .125 Mabswid .5 .03023 m .50375 .03023 L s .5 .06046 m .50375 .06046 L s .5 .09068 m .50375 .09068 L s .5 .12091 m .50375 .12091 L s .5 .18137 m .50375 .18137 L s .5 .2116 m .50375 .2116 L s .5 .24182 m .50375 .24182 L s .5 .27205 m .50375 .27205 L s .5 .33251 m .50375 .33251 L s .5 .36274 m .50375 .36274 L s .5 .39296 m .50375 .39296 L s .5 .42319 m .50375 .42319 L s .5 .48365 m .50375 .48365 L s .5 .51388 m .50375 .51388 L s .5 .5441 m .50375 .5441 L s .5 .57433 m .50375 .57433 L s .25 Mabswid .5 0 m .5 .61803 L s gsave .5 .64303 -65.5312 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 64.250 12.000 moveto %%IncludeResource: font Times-Italic %%IncludeFont: Times-Italic /Times-Italic findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (f) show 68.062 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 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 2 Mabswid .02381 0 m .02605 0 L .02846 0 L .03279 0 L .03543 0 L .03793 0 L .04018 0 L .04262 0 L .045 0 L .04762 0 L .05036 0 L .05293 0 L .05751 0 L .06009 0 L .06244 0 L .06504 0 L .0678 0 L .07041 0 L .07277 0 L .07556 0 L .07858 0 L .08143 0 L .08405 0 L .08881 0 L .09151 0 L .09403 0 L .09912 0 L .10198 0 L .10458 0 L .10939 0 L .11181 0 L .11449 0 L .11677 0 L .11922 0 L .12366 0 L .12618 0 L .12894 0 L .13163 0 L .1345 0 L .13719 0 L .13964 0 L .14443 0 L .14673 0 L .14927 0 L .15175 0 L .15441 0 L .15916 0 L .16361 0 L .16628 0 L .16918 0 L Mistroke .17448 1e-05 L .17963 1e-05 L .18441 1e-05 L .1895 2e-05 L .195 3e-05 L .2002 4e-05 L .20497 5e-05 L .20929 7e-05 L .21403 9e-05 L .21898 .00012 L .22359 .00016 L .22861 .00022 L .23407 .0003 L .23922 .00041 L .24398 .00053 L .24827 .00067 L .25299 .00086 L .25792 .00111 L .26255 .00141 L .27241 .00231 L .27736 .00294 L .28277 .00379 L .28742 .0047 L .29237 .00588 L .30129 .00868 L .30664 .01087 L .31156 .0133 L .32268 .02058 L .33248 .02959 L .34284 .04247 L .3523 .05789 L .36245 .07902 L .38078 .13099 L .40199 .21489 L .42211 .31424 L .4403 .4112 L .4604 .50952 L .46991 .54707 L .48016 .57799 L .4829 .58433 L .4858 .59005 L .48848 .59444 L .49098 .59772 L .49333 .60009 L .49464 .60111 L .49584 .60184 L .49715 .60244 L .49789 .60267 L .49857 .60283 L .49983 .60296 L Mistroke .50046 .60295 L .50115 .60288 L .50245 .60257 L .50364 .6021 L .50496 .60137 L .50636 .60035 L .50895 .59779 L .5113 .59474 L .51608 .58644 L .52058 .57614 L .53061 .54521 L .54146 .50129 L .58249 .29032 L .60292 .19324 L .62199 .1219 L .64033 .0727 L .65061 .05272 L .65998 .03857 L .66991 .02713 L .67908 .01924 L .68916 .01292 L .69424 .01048 L .69985 .00826 L .70477 .00667 L .70995 .0053 L .71921 .00346 L .72438 .0027 L .72918 .00214 L .73441 .00165 L .74007 .00123 L .74555 .00093 L .75061 .00071 L .75544 .00055 L .76055 .00041 L .76512 .00032 L .7694 .00025 L .7741 .00019 L .77914 .00014 L .7844 .0001 L .78928 8e-05 L .79463 5e-05 L .80033 4e-05 L .80577 3e-05 L .81088 2e-05 L .81548 1e-05 L .82043 1e-05 L .82537 1e-05 L .82997 1e-05 L .83502 0 L .83785 0 L Mistroke .84043 0 L .84562 0 L .84855 0 L .85126 0 L .85624 0 L .85875 0 L .86151 0 L .86399 0 L .8667 0 L .86909 0 L .87164 0 L .87416 0 L .87645 0 L .88091 0 L .8836 0 L .88607 0 L .88873 0 L .89166 0 L .89658 0 L .89927 0 L .90179 0 L .90679 0 L .90961 0 L .91223 0 L .91703 0 L .91945 0 L .92213 0 L .92673 0 L .92932 0 L .9317 0 L .93607 0 L .93827 0 L .94068 0 L .9429 0 L .94531 0 L .94743 0 L .94967 0 L .9517 0 L .95393 0 L .95785 0 L .96023 0 L .96241 0 L .96476 0 L .96732 0 L .96968 0 L .97187 0 L .97619 0 L Mfstroke .5 Mabswid .02381 .00068 m .03279 .00086 L .04262 .0011 L .06244 .00176 L .07298 .00225 L .08426 .00291 L .10458 .00453 L .1148 .00561 L .12589 .00703 L .14603 .01042 L .16586 .01504 L .18689 .02167 L .20643 .02979 L .22473 .0394 L .2635 .06713 L .30472 .10828 L .34443 .1574 L .38262 .20825 L .40394 .23532 L .42327 .25739 L .44365 .27684 L .4624 .29025 L .47176 .29509 L .47653 .29706 L .48165 .29877 L .48433 .2995 L .48678 .30007 L .48927 .30055 L .49157 .30091 L .49425 .30121 L .49566 .30133 L .49639 .30138 L .49717 .30142 L .49847 .30146 L .49969 .30148 L .50039 .30148 L .50114 .30147 L .50177 .30146 L .50247 .30143 L .50379 .30136 L .505 .30128 L .50634 .30116 L .50775 .30099 L .51013 .30065 L .51274 .30017 L .51756 .29899 L .52208 .29756 L .53218 .29321 L .54303 .28686 L .5642 .26989 L Mistroke .5843 .2491 L .62255 .20142 L .66325 .14738 L .70243 .10031 L .72244 .07983 L .74406 .06089 L .78418 .03447 L .80416 .02514 L .82524 .0176 L .84491 .01235 L .86326 .00871 L .88365 .00579 L .90222 .00391 L .91216 .00315 L .92134 .00256 L .93207 .002 L .94212 .00158 L .95833 .00107 L .96693 .00086 L .97619 .00068 L Mfstroke [ 7 5 ] 0 Mabsdash .02381 .01342 m .06244 .02045 L .10458 .0311 L .14415 .04434 L .18221 .06021 L .22272 .08029 L .26171 .10206 L .30316 .12657 L .34309 .14981 L .3815 .16997 L .40095 .17878 L .42237 .18704 L .44268 .19326 L .45178 .19549 L .46172 .1975 L .4671 .19841 L .4721 .19913 L .47727 .19975 L .48196 .20021 L .48658 .20056 L .4887 .20068 L .49093 .20079 L .49332 .20088 L .49438 .20091 L .49552 .20094 L .49675 .20096 L .49789 .20098 L .49859 .20098 L .49925 .20099 L .50049 .20099 L .50163 .20098 L .50286 .20097 L .50401 .20095 L .50508 .20093 L .50754 .20085 L .51014 .20074 L .51268 .2006 L .51504 .20045 L .5204 .19999 L .5293 .19894 L .53882 .1974 L .54906 .1953 L .56016 .19249 L .58032 .18609 L .62123 .16865 L .66064 .14771 L .69852 .12557 L .73886 .10172 L .77769 .08007 L .81897 .05968 L Mistroke .85873 .04327 L .89697 .03064 L .93767 .02043 L .97619 .01342 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{294, 180.25}, ImageMargins->{{42.25, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggoo00<007ooOol077oo10007Goo00@007ooOol003Mo o`00:Woo0`007Goo00<007oo00008Goo00<007ooOol0>goo00<007ooOol06goo00<007oo00007Woo 0`00>7oo000ZOol00`00Oomoo`0NOol2000NOol01000Oomoo`00>Woo00@007ooOol001ioo`8001io o`03001oogoo03Qoo`00:goo0P007goo00<007ooOol07Goo0P00?7oo0P0087oo00<007ooOol07Goo 0P00>7oo003oOolWOol00?moobMoo`00ogoo9goo003oOolWoo000_Ool3000FOol00`00Oomoo`0GOol3000VOol00`00Oomoo`0ROol3000K Ool2000DOol20010Ool0039oo`03001oogoo01Aoo`8001Qoo`<002Eoo`03001oogoo029oo`8001Yo o`8001Aoo`80049oo`00Bgoo00<007ooOol05Woo0P009Goo00<007ooOol08Goo0`006Goo00<007oo Ool04Woo0P00A7oo000gOol3000BOol2000FOol3000TOol00`00Oomoo`0QOol2000HOol2000COol2 0016Ool003Yoo`<0015oo`03001oogoo01Aoo`8002Aoo`03001oogoo021oo`<001Ioo`8005eoo`00 ?Goo00<007ooOol03goo0P0057oo0`008goo00<007ooOol087oo0P005Woo00<007ooOol047oo0P00 Bgoo001AOol00`00Oomoo`0BOol2000SOol2000POol3000DOol2000AOol2001=Ool0049oo`8000io o`03001oogoo015oo`<0029oo`03001oogoo01moo`8001Aoo`03001oogoo00moo`8004moo`00A7oo 00<007ooOol037oo00<007ooOol04Goo0`008Goo00<007ooOol07goo0P004goo00<007ooOol03Goo 0`00DGoo0015Ool2000=Ool00`00Oomoo`0AOol2000QOol00`00Oomoo`0NOol3000BOol00`00Oomo o`0Ool00`00Oomoo`03Ool2001UOol005Moo`8000Aoo`03001o ogoo00]oo`<001eoo`03001oogoo01Yoo`8000ioo`03001oogoo009oo`8006Moo`00FGoo0P000goo 00<007ooOol02goo0P007Goo0P006Woo0`003Woo00<007ooOol00P00JGoo001KOol01@00Oomoogoo 000037oo0P007Goo00<007ooOol06Goo0P003Woo00@007ooOol006]oo`00G7oo0P000Woo00<007oo Ool02Goo0`0077oo00<007ooOol06Goo0P003Goo00<007ooOol0KGoo001QOol00`00Oomoo`09Ool2 000LOol00`00Oomoo`0HOol3000Ool40004Ool20005Ool00`00Oomoo`1h Ool006aoo`04001oogooOol20008Ool3000>Ool00`00Oomoo`0Ool30027Ool007Moo`<0015oo`03001oogoo00ioo`8008Qoo`00N7oo0P004Goo 0P003goo0P00R7oo001hOol2000AOol00`00Oomoo`0>Ool20028Ool007Qoo`80015oo`03001oogoo 00ioo`8008Qoo`00N7oo0P004Goo00<007ooOol03Goo0`00R7oo001hOol3000@Ool00`00Oomoo`0= Ool20029Ool007Uoo`80011oo`03001oogoo00eoo`8008Uoo`00NGoo0P0047oo00<007ooOol03Goo 0P00RGoo001iOol2000@Ool2000>Ool20029Ool007Uoo`<000moo`03001oogoo00aoo`<008Uoo`00 NWoo0P003goo00<007ooOol037oo0P00RWoo001jOol2000?Ool00`00Oomoo`0Ool00`00Oomoo`0;Ool3002:Ool007]oo`8000ioo`03001oogoo00]oo`8008]oo`00Ngoo0P00 3Woo0P0037oo0P00Rgoo001kOol2000>Ool00`00Oomoo`0;Ool2002;Ool007]oo`8000ioo`03001o ogoo00]oo`8008]oo`00Ngoo0`003Goo00<007ooOol02Woo0`00Rgoo001lOol2000=Ool00`00Oomo o`0:Ool2002Ool007]oo`03001oogoo008000Ioo`03001oogoo009oo`03 001oogoo00Qoo`8008ioo`00Ngoo00<007ooOol00P001Goo00<007ooOol00goo00<007ooOol027oo 0P00SWoo001kOol00`00Oomoo`020003Ool01000Oomoo`0017oo00<007ooOol027oo0P00SWoo001l Ool40004Ool20005Ool00`00Oomoo`07Ool3002>Ool007ioo`<000Yoo`03001oogoo00Moo`8008mo o`00Ogoo0P002Woo00<007ooOol01goo0P00Sgoo001oOol2000:Ool20008Ool2002?Ool007moo`80 00Yoo`03001oogoo00Moo`8008moo`00Ogoo0P002Woo00<007ooOol01Woo0`00Sgoo001oOol2000: Ool00`00Oomoo`06Ool2002@Ool007moo`<000Uoo`03001oogoo00Ioo`80091oo`00P7oo0P002Goo 00<007ooOol01Woo0P00T7oo0020Ool20009Ool00`00Oomoo`06Ool2002@Ool0081oo`8000Uoo`03 001oogoo00Eoo`<0091oo`00P7oo0`0027oo0P001Woo0P00TGoo0021Ool20008Ool00`00Oomoo`05 Ool2002AOol0085oo`8000Qoo`03001oogoo00Aoo`<0095oo`00PGoo0`001goo00<007ooOol017oo 0P00TWoo0022Ool20007Ool00`00Oomoo`04Ool2002BOol0089oo`8000Moo`03001oogoo00Aoo`80 099oo`00PWoo0`001Woo00<007ooOol00goo0`00TWoo0023Ool20006Ool20004Ool2002COol008=o o`8000Ioo`03001oogoo00=oo`8009=oo`00Pgoo0P001Woo00<007ooOol00goo0P00Tgoo0023Ool3 0005Ool00`00Oomoo`02Ool3002COol008Aoo`8000Eoo`03001oogoo009oo`8009Aoo`00Q7oo0P00 1Goo00<007ooOol00Woo0P00U7oo0024Ool20005Ool01000Oomoogoo0`00U7oo0024Ool30004Ool2 0002Ool2002EOol008Eoo`8000Aoo`04001oogooOol2002EOol008Eoo`<000=oo`03001oogoo00<0 09Eoo`00QWoo0P000goo00<007ooOol00P00UWoo0026Ool20003Ool00`00Oomoo`02002FOol008Io o`<0009oo`03001oo`00008009Ioo`00O7oo0P000goo00H007ooOomoo`00Ool8002GOol007]oo`04 001oogoo0006Ool00`00Oomoo`06002HOol007]oo`04001oogoo0004Ool40002Ool4002IOol007]o o`04001oogoo0004Ool00`00Ool00005Ool00`00Oomoo`2HOol007]oo`04001oogoo0005Ool20005 Ool00`00Oomoo`2HOol007aoo`8000Moo`03001oogoo00=oo`03001oogoo09Qoo`00ogoo9goo003o OolWOol00?moobMoo`00ogoo9goo003oOolWOol00?moobMoo`00R7oo0P00W7oo002:Ool00`00Oomo o`2IOol008Yoo`03001oogoo09Uoo`00RWoo00<007ooOol0VGoo002;Ool00`00Oomoo`2HOol008Yo o`<009Uoo`00Rgoo00<007ooOol0V7oo002"], ImageRangeCache->{{{136.375, 429.375}, {617.063, 437.813}} -> {-16.849, \ 1.16896, 0.0611023, 0.00275821}}] }, Open ]], Cell[TextData[{ StyleBox["Fig. ", FontWeight->"Bold"], StyleBox[ CounterBox["Figure"], FontWeight->"Bold"], StyleBox[": ", FontWeight->"Bold"], " The pdf of a Normal random variable, when ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\ = \ 0\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Sigma] = 1, 2\)]], " and ", Cell[BoxData[ \(TraditionalForm\`3\)]] }], "FigureLabel"], Cell[TextData[{ "It is well known that ", Cell[BoxData[ \(TraditionalForm\`E[X]\ = \ \[Mu]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`Var(X) = \ \[Sigma]\^2\)]], ", as we can easily verify:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Expect[x, \ f]\)], "Input"], Cell[BoxData[ \(\[Mu]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Var[x, \ f]\)], "Input"], Cell[BoxData[ \(\[Sigma]\^2\)], "Output"] }, Open ]], Cell[TextData[{ "Because the ", StyleBox["PDF", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " package is completely general in its design, we can just as easily solve \ problems that are both less well-known and more 'difficult', such as finding \ ", Cell[BoxData[ \(TraditionalForm\`\(\(Var(X\^2)\)\(\ \)\)\)]], ": " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Var[x\^2, \ f]\)], "Input"], Cell[BoxData[ \(2\ \((2\ \[Mu]\^2\ \[Sigma]\^2 + \[Sigma]\^4)\)\)], "Output"] }, Open ]], Cell[TextData[{ "Assumptions technology is a very important addition to ", StyleBox["Mathematica", FontSlant->"Italic"], ". In order for it to work, one should enter as much information about \ parameters as possible: not only will the resulting answer be much neater, it \ may also be obtained faster, and it may make it possible to solve problems \ that could not otherwise be solved. Here are some examples of Assumptions \ statements:" }], "IndentText"], Cell[BoxData[ \(\(\({\[Alpha]\ > 1, \ \ \ \[Beta]\ \[Element] \ Integers, \ \ \ \(-\[Infinity]\) < \ \[Gamma]\ < \[Pi], \ \ \ \ \[Delta]\ \[Element] \ Reals, \ \ \[Theta]\ > \ 0}\)\(\ \ \ \)\)\)], "Input"], Cell[TextData[{ "The ", StyleBox["PDF", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " package implements assumptions technology in a ", StyleBox["distribution", FontSlant->"Italic"], "-specific manner. That is, the assumptions are attached to the density ", Cell[BoxData[ \(TraditionalForm\`f(x; \[Theta])\)]], ", and not to the parameter \[Theta]. Because the assumptions are not \ global, it is important to realise that the assumptions will only be \ automatically invoked when using the suite of ", StyleBox["PDF", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " package functions. By contrast, ", StyleBox["Mathematica", FontSlant->"Italic"], "'s inbuilt functions such as the derivative function, ", StyleBox["D[", FontFamily->"Courier"], StyleBox["f", FontSlant->"Italic"], ", ", StyleBox["x", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier"], ", will not automatically assume that ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\ > 0\)]], " when presenting their solution." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["2.3\tDiscrete random variables", "Subsubsection"], Cell[TextData[{ "The ", StyleBox["PDF", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " toolset automatically handles discrete random variables in the same way. \ The only difference is that, when we define the density, we add a flag to \ tell ", StyleBox["Mathematica", FontSlant->"Italic"], " that the random variable is ", StyleBox["{Discrete}", FontFamily->"Courier"], ". For instance: " }], "Text"], Cell[TextData[{ "Let the discrete random variable ", Cell[BoxData[ \(TraditionalForm\`X\)]], " have probability mass function (pmf) ", Cell[BoxData[ FormBox[ RowBox[{\(f(x)\), " ", "=", " ", RowBox[{ TagBox[ RowBox[{"(", GridBox[{ { TagBox[\(r + x - 1\), Identity, Editable->True]}, { TagBox["x", Identity, Editable->True]} }], ")"}], InterpretTemplate[ Binomial[ #, #2]&], Editable->False], " ", \(p\^r\), " ", \(\((1 - p)\)\^x\)}]}], TraditionalForm]]], ", where ", Cell[BoxData[ \(TraditionalForm\`x = 0, \ 1, \ 2, \ ... \)]], ". Here parameter ", Cell[BoxData[ \(TraditionalForm\`\(\(p\)\(\ \)\)\)]], " is the probability of success, while parameter ", Cell[BoxData[ \(TraditionalForm\`r\)]], " is a positive integer. In ", StyleBox["Mathematica", FontSlant->"Italic"], ", we enter this as: " }], "Text"], Cell[BoxData[{ \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(f\ \ = \ \ Binomial[r + x - 1, x]\ p\^r\ \((1 - p)\)\^x;\)\), "\n", \(\(domain[ f]\ = \ {x, 0, \[Infinity]}\ \ && \ {Discrete}\ \ && \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {0 < p < 1, \ \ r\ > 0, \ r \[Element] Integers}\ ;\)\)}], "Input", GeneratedCell->False], Cell[TextData[{ "This is known as the Pascal distribution. Here is a plot of ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(PlotDensity[ f /. \ {p \[Rule] \ 1\/2, \ r \[Rule] \ 10\ }];\)\)], "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 -1.56593e-17 0.0353607 0 6.47038 [ [.1768 -0.0125 -4.25 -11 ] [.1768 -0.0125 4.25 0 ] [.35361 -0.0125 -6.5 -11 ] [.35361 -0.0125 6.5 0 ] [.53041 -0.0125 -6.5 -11 ] [.53041 -0.0125 6.5 0 ] [.70721 -0.0125 -6.5 -11 ] [.70721 -0.0125 6.5 0 ] [.88402 -0.0125 -6.5 -11 ] [.88402 -0.0125 6.5 0 ] [1.025 0 0 -5.5 ] [1.025 0 8.25 5.5 ] [-0.0125 .12941 -19.75 -5.5 ] [-0.0125 .12941 0 5.5 ] [-0.0125 .25882 -19.75 -5.5 ] [-0.0125 .25882 0 5.5 ] [-0.0125 .38822 -19.75 -5.5 ] [-0.0125 .38822 0 5.5 ] [-0.0125 .51763 -19.75 -5.5 ] [-0.0125 .51763 0 5.5 ] [0 .64303 -4.53125 0 ] [0 .64303 4.53125 11 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .1768 0 m .1768 .00625 L s gsave .1768 -0.0125 -65.25 -15 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (5) show 1.000 setlinewidth grestore .35361 0 m .35361 .00625 L s gsave .35361 -0.0125 -67.5 -15 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (10) show 1.000 setlinewidth grestore .53041 0 m .53041 .00625 L s gsave .53041 -0.0125 -67.5 -15 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (15) show 1.000 setlinewidth grestore .70721 0 m .70721 .00625 L s gsave .70721 -0.0125 -67.5 -15 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (20) show 1.000 setlinewidth grestore .88402 0 m .88402 .00625 L s gsave .88402 -0.0125 -67.5 -15 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (25) show 1.000 setlinewidth grestore .125 Mabswid .03536 0 m .03536 .00375 L s .07072 0 m .07072 .00375 L s .10608 0 m .10608 .00375 L s .14144 0 m .14144 .00375 L s .21216 0 m .21216 .00375 L s .24752 0 m .24752 .00375 L s .28289 0 m .28289 .00375 L s .31825 0 m .31825 .00375 L s .38897 0 m .38897 .00375 L s .42433 0 m .42433 .00375 L s .45969 0 m .45969 .00375 L s .49505 0 m .49505 .00375 L s .56577 0 m .56577 .00375 L s .60113 0 m .60113 .00375 L s .63649 0 m .63649 .00375 L s .67185 0 m .67185 .00375 L s .74257 0 m .74257 .00375 L s .77793 0 m .77793 .00375 L s .8133 0 m .8133 .00375 L s .84866 0 m .84866 .00375 L s .91938 0 m .91938 .00375 L s .95474 0 m .95474 .00375 L s .9901 0 m .9901 .00375 L s .25 Mabswid 0 0 m 1 0 L s gsave 1.025 0 -61 -9.5 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 63.250 12.000 moveto %%IncludeResource: font Times-Italic %%IncludeFont: Times-Italic /Times-Italic findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (x) show 67.250 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 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 .12941 m .00625 .12941 L s gsave -0.0125 .12941 -80.75 -9.5 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (0.02) show 1.000 setlinewidth grestore 0 .25882 m .00625 .25882 L s gsave -0.0125 .25882 -80.75 -9.5 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (0.04) show 1.000 setlinewidth grestore 0 .38822 m .00625 .38822 L s gsave -0.0125 .38822 -80.75 -9.5 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (0.06) show 1.000 setlinewidth grestore 0 .51763 m .00625 .51763 L s gsave -0.0125 .51763 -80.75 -9.5 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (0.08) show 1.000 setlinewidth grestore .125 Mabswid 0 .03235 m .00375 .03235 L s 0 .0647 m .00375 .0647 L s 0 .09706 m .00375 .09706 L s 0 .16176 m .00375 .16176 L s 0 .19411 m .00375 .19411 L s 0 .22646 m .00375 .22646 L s 0 .29117 m .00375 .29117 L s 0 .32352 m .00375 .32352 L s 0 .35587 m .00375 .35587 L s 0 .42058 m .00375 .42058 L s 0 .45293 m .00375 .45293 L s 0 .48528 m .00375 .48528 L s 0 .54998 m .00375 .54998 L s 0 .58233 m .00375 .58233 L s 0 .61469 m .00375 .61469 L s .25 Mabswid 0 0 m 0 .61803 L s gsave 0 .64303 -65.5312 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 19 translate 1 -1 scale 63.000 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 64.250 12.000 moveto %%IncludeResource: font Times-Italic %%IncludeFont: Times-Italic /Times-Italic findfont 9.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (f) show 68.062 12.000 moveto %%IncludeResource: font Times-Roman %%IncludeFont: Times-Roman /Times-Roman findfont 9.000 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 .7 g .2 Mabswid 0 0 m 0 .00632 L s .03536 0 m .03536 .03159 L s .07072 0 m .07072 .08688 L s .10608 0 m .10608 .17377 L s .14144 0 m .14144 .28237 L s .1768 0 m .1768 .39532 L s .21216 0 m .21216 .49414 L s .24752 0 m .24752 .56474 L s .28289 0 m .28289 .60003 L s .31825 0 m .31825 .60003 L s .35361 0 m .35361 .57003 L s .38897 0 m .38897 .51821 L s .42433 0 m .42433 .45343 L s .45969 0 m .45969 .38367 L s .49505 0 m .49505 .31516 L s .53041 0 m .53041 .25213 L s .56577 0 m .56577 .19698 L s .60113 0 m .60113 .15063 L s .63649 0 m .63649 .11297 L s .67185 0 m .67185 .08324 L s .70721 0 m .70721 .06035 L s .74257 0 m .74257 .04311 L s .77793 0 m .77793 .03037 L s .8133 0 m .8133 .02113 L s .84866 0 m .84866 .01453 L s .88402 0 m .88402 .00988 L s .91938 0 m .91938 .00665 L s .95474 0 m .95474 .00443 L s .9901 0 m .9901 .00293 L s 0 .1 1 r 5 Mabswid 0 .00632 Mdot .03536 .03159 Mdot .07072 .08688 Mdot .10608 .17377 Mdot .14144 .28237 Mdot .1768 .39532 Mdot .21216 .49414 Mdot .24752 .56474 Mdot .28289 .60003 Mdot .31825 .60003 Mdot .35361 .57003 Mdot .38897 .51821 Mdot .42433 .45343 Mdot .45969 .38367 Mdot .49505 .31516 Mdot .53041 .25213 Mdot .56577 .19698 Mdot .60113 .15063 Mdot .63649 .11297 Mdot .67185 .08324 Mdot .70721 .06035 Mdot .74257 .04311 Mdot .77793 .03037 Mdot .8133 .02113 Mdot .84866 .01453 Mdot .88402 .00988 Mdot .91938 .00665 Mdot .95474 .00443 Mdot .9901 .00293 Mdot % End of Graphics MathPictureEnd \ \>"], "Graphics", CellMargins->{{43, Inherited}, {Inherited, Inherited}}, ImageSize->{386.25, 238.125}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgWoo001KOol00`00Oomoo`0cOol01000Oomoo`000Woo00<007ooOol0 Ool0021oo`80025oo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qo o`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qo o`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qo o`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo08ioo`0087oo00<007ooOol087oo00=JeWoo Ool027oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWoo Ool027oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWoo Ool027oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWoo Ool0SWoo000POol00`00Oomoo`0POol00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool0 0e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool0 0e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool0 0e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`2>Ool0021oo`03001oogoo021oo`03F]Ioogoo 00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo 00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo 00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo 08ioo`0087oo00<007ooOol087oo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=J eWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=J eWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=J eWooOol02Goo00=JeWooOol027oo0`1oSgoo000POol00`00Oomoo`0POol00e[FOomoo`08Ool00e[F Oomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[F Oomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[F Oomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`07Ool507n>Ool0021oo`03001o ogoo021oo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Io ogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Io ogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Io ogoo00Moo`D0Ohioo`0087oo00<007ooOol087oo00=JeWooOol027oo00=JeWooOol02Goo00=JeWoo Ool027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWoo Ool027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWoo Ool027oo00=JeWooOol02Goo00=JeWooOol01goo1@1oSWoo000POol00`00Oomoo`0POol00e[FOomo o`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomo o`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomo o`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool307n?Ool0 00]oo`8000=oo`03001oogoo0080009oo`@000Eoo`03001oogoo021oo`03F]Ioogoo00Qoo`03F]Io ogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Io ogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Io ogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo09Yoo`002Woo00@007ooOol000Ao o`04001oogoo0002Ool00`00Oomoo`05Ool00`00Oomoo`0POol00e[FOomoo`08Ool00e[FOomoo`09 Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09 Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09 Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`2JOol000Yoo`04001oogoo0004Ool01000 Oomoo`000goo00<007ooOol017oo0`0087oo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol0 27oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol0 27oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol0 27oo00=JeWooOol02Goo00=JeWooOol0VWoo000:Ool01000Oomoo`0017oo00@007ooOol000Aoo`03 001oogoo00=oo`03001oogoo021oo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03 F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03 F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03 F]Ioogoo00Uoo`03F]Ioogoo09Yoo`002Woo00@007ooOol000Aoo`06001oogoo001oo`000Woo00<0 07ooOol00goo00<007ooOol087oo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=J eWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=J eWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=J eWooOol02Goo00=JeWooOol0VWoo000;Ool20006Ool20003Ool20006Ool00`00Oomoo`0POol00e[F Oomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[F Oomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[F Oomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`2JOol0021o o`03001oogoo021oo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uo o`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uo o`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uo o`03F]Ioogoo09Yoo`0087oo00<007ooOol087oo00=JeWooOol027oo00=JeWooOol02Goo00=JeWoo Ool027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWoo Ool027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWoo Ool027oo00=JeWooOol027oo0`1oVgoo000POol00`00Oomoo`0POol00e[FOomoo`08Ool00e[FOomo o`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomo o`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomo o`09Ool00e[FOomoo`08Ool00e[FOomoo`07Ool507nJOol0021oo`03001oogoo021oo`03F]Ioogoo 00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo 00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo 00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Moo`D0OiYoo`0087oo00<007ooOol0 87oo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol0 2Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol0 2Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol01goo1@1oVWoo000P Ool00`00Oomoo`0POol00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09 Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09 Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`08 Ool307nKOol0021oo`03001oogoo021oo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qo o`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qo o`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qo o`03F]Ioogoo0:Ioo`0087oo0P008Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo 00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo 00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo 00=JeWooOol0YWoo000POol00`00Oomoo`0POol00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomo o`08Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomo o`08Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomo o`08Ool00e[FOomoo`2VOol0021oo`03001oogoo01moo`<0O`Uoo`03F]Ioogoo00Uoo`03F]Ioogoo 00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo 00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo 00Qoo`03F]Ioogoo0:Ioo`0087oo00<007ooOol07Woo1@1o27oo00=JeWooOol02Goo00=JeWooOol0 27oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol0 27oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol0 27oo00=JeWooOol0YWoo000POol00`00Oomoo`0NOol507l8Ool00e[FOomoo`09Ool00e[FOomoo`08 Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08 Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08 Ool00e[FOomoo`2VOol0021oo`03001oogoo01ioo`D0O`Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qo o`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qo o`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qo o`03F]Ioogoo0:Ioo`0087oo00<007ooOol07goo0`1o2Goo00=JeWooOol02Goo00=JeWooOol027oo 00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo 00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo 00=JeWooOol0YWoo000POol00`00Oomoo`0[Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomo o`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomo o`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomo o`2VOol0021oo`03001oogoo02]oo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03 F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03 F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo0:Ioo`00 87oo00<007ooOol0:goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol0 2Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol0 2Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol01goo0`1oYgoo000POol2000/Ool00e[F Oomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[F Oomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[F Oomoo`09Ool00e[FOomoo`06Ool507nVOol0021oo`03001oogoo02]oo`03F]Ioogoo00Uoo`03F]Io ogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Io ogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Io ogoo00Ioo`D0OjIoo`0087oo00<007ooOol0:goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWoo Ool02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWoo Ool02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol01Woo1@1oYWoo 000POol00`00Oomoo`0[Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomo o`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomo o`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`07Ool307nWOol0021oo`03001oogoo 02]oo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo 00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo 00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo0;5oo`0087oo00<007ooOol0:goo00=JeWooOol02Goo00=J eWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=J eWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=J eWooOol0/Goo000POol00`00Oomoo`0[Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09 Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09 Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`2aOol0021oo`03001o ogoo02]oo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Io ogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Io ogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo0;5oo`0087oo00<007ooOol0:goo00=JeWooOol02Goo 00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo 00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo 00=JeWooOol0/Goo000POol00`00Oomoo`0[Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomo o`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomo o`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`2aOol0021oo`03 001oogoo02]oo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03 F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03 F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo0;5oo`0087oo0P00;7oo00=JeWooOol02Goo00=J eWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=J eWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=J eWooOol0/Goo000POol00`00Oomoo`0[Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09 Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09 Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`2aOol0021oo`03001o ogoo02]oo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Io ogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Io ogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo0;5oo`0087oo00<007ooOol0:goo00=JeWooOol02Goo 00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo 00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo 00=JeWooOol0/Goo000POol00`00Oomoo`0[Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomo o`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomo o`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`2aOol0021oo`03 001oogoo02]oo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03 F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03 F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo0;5oo`0087oo00<007ooOol0:goo00=JeWooOol0 2Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol0 2Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol0 27oo0`1o/Woo000POol00`00Oomoo`0[Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09 Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09 Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`07Ool507naOol0021oo`03001oogoo02]o o`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qo o`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qo o`03F]Ioogoo00Moo`D0Ok5oo`002goo0P000goo00<007ooOol00P0017oo00<007ooOol017oo00<0 07ooOol0:goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=J eWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=J eWooOol027oo00=JeWooOol01goo1@1o/Goo000:Ool01000Oomoo`0017oo00@007ooOol000=oo`03 001oogoo00Aoo`<002]oo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo 00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo 00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Qoo`<0Ok9oo`002Woo00@007ooOol000Aoo`05001oogoo 001oo`040005Ool00`00Oomoo`0[Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool0 0e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool0 0e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`2mOol000Yoo`04001oogoo0004Ool02000Oomo o`00Ool007oo0006Ool00`00Oomoo`0[Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09 Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09 Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`2mOol000Yoo`04001oogoo0004Ool01000 Oomoo`000Woo0P001Woo00<007ooOol0:goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol0 2Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol0 2Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol0_Goo000;Ool20006Ool20004Ool00`00 Oomoo`04Ool00`00Oomoo`0[Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[F Oomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[F Oomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`2mOol0021oo`03001oogoo02]oo`03F]Ioogoo00Uo o`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uo o`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo0;eo o`0087oo00<007ooOol0:Woo0`1o2Woo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo 00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo 00=JeWooOol027oo00=JeWooOol0_Goo000POol00`00Oomoo`0YOol507l9Ool00e[FOomoo`08Ool0 0e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool0 0e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`2mOol0021oo`03001oogoo 02Uoo`D0O`Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03 F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03 F]Ioogoo0;eoo`0087oo00<007ooOol0:Goo1@1o2Goo00=JeWooOol027oo00=JeWooOol02Goo00=J eWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=J eWooOol02Goo00=JeWooOol027oo00=JeWooOol0_Goo000POol00`00Oomoo`0ZOol307l:Ool00e[F Oomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[F Oomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`2mOol0021o o`8003Qoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Io ogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Io ogoo0;eoo`0087oo00<007ooOol0=goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo 00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo 00=JeWooOol027oo00=JeWooOol0_Goo000POol00`00Oomoo`0gOol00e[FOomoo`08Ool00e[FOomo o`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomo o`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`2mOol0021oo`03001oogoo03Moo`03 F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03 F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo0;eoo`00 87oo00<007ooOol0=goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol0 27oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol0 27oo00=JeWooOol0_Goo000POol00`00Oomoo`0gOol00e[FOomoo`08Ool00e[FOomoo`09Ool00e[F Oomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[F Oomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`2mOol0021oo`03001oogoo03Moo`03F]Ioogoo00Qo o`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qo o`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Moo`<0Okioo`0087oo00<007ooOol0=goo 00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo 00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol01Woo1@1o_Goo000POol0 0`00Oomoo`0gOol00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool0 0e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`06Ool5 07nmOol0021oo`03001oogoo03Moo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03 F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03 F]Ioogoo00Ioo`D0Okeoo`0087oo00<007ooOol0=goo00=JeWooOol027oo00=JeWooOol02Goo00=J eWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=J eWooOol02Goo00=JeWooOol01goo0`1o_Woo000POol2000hOol00e[FOomoo`08Ool00e[FOomoo`09 Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09 Ool00e[FOomoo`09Ool00e[FOomoo`38Ool0021oo`03001oogoo03Moo`03F]Ioogoo00Qoo`03F]Io ogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Io ogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo01oo`0087oo00<007ooOol0@Woo00=JeWooOol02Goo00=J eWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol0h7oo000P Ool00`00Oomoo`12Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09 Ool00e[FOomoo`08Ool00e[FOomoo`3POol0021oo`03001oogoo049oo`03F]Ioogoo00Uoo`03F]Io ogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo0>1oo`0087oo 00<007ooOol0@Woo00=JeWooOol02Goo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo 00=JeWooOol027oo00=JeWooOol0h7oo000POol00`00Oomoo`12Ool00e[FOomoo`09Ool00e[FOomo o`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`3POol0021oo`03 001oogoo049oo`03F]Ioogoo00Uoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03 F]Ioogoo00Qoo`03F]Ioogoo0>1oo`0087oo00<007ooOol0@Woo00=JeWooOol02Goo00=JeWooOol0 2Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol0h7oo000POol20013 Ool00e[FOomoo`09Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`08 Ool00e[FOomoo`3POol0021oo`03001oogoo045oo`<0O`Yoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qo o`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo0>1oo`0087oo00<007ooOol0@7oo1@1o2Goo 00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol0h7oo 000POol00`00Oomoo`10Ool507l9Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool0 0e[FOomoo`08Ool00e[FOomoo`3POol0021oo`03001oogoo041oo`D0O`Uoo`03F]Ioogoo00Uoo`03 F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo0>1oo`0087oo00<007ooOol0 @Goo0`1o2Woo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=J eWooOol0h7oo000POol00`00Oomoo`1>Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09 Ool00e[FOomoo`08Ool00e[FOomoo`3POol0021oo`03001oogoo04ioo`03F]Ioogoo00Uoo`03F]Io ogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo0>1oo`0087oo00<007ooOol0CWoo 00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol0h7oo 000;Ool20003Ool00`00Oomoo`020003Ool20006Ool00`00Oomoo`1>Ool00e[FOomoo`09Ool00e[F Oomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`07Ool307oQOol000Yoo`04001oogoo0004Ool01P00 Oomoo`00Ool0009oo`03001oogoo00=oo`<004ioo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Io ogoo00Uoo`03F]Ioogoo00Ioo`D0On1oo`002Woo00@007ooOol000Aoo`06001oogoo001oo`000Woo 00<007ooOol00goo00<007ooOol0CWoo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo 00=JeWooOol01Woo1@1oh7oo000:Ool01000Oomoo`0017oo00@007ooOol0009oo`8000Ioo`03001o ogoo04ioo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo00Ioo`D0On1o o`002Woo00@007ooOol000Aoo`06001oogoo001oo`000Woo00<007ooOol00goo00<007ooOol0CWoo 00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol01goo0`1ohGoo000;Ool2 0006Ool20003Ool20006Ool00`00Oomoo`1>Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomo o`09Ool00e[FOomoo`3[Ool0021oo`03001oogoo04ioo`03F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03 F]Ioogoo00Uoo`03F]Ioogoo0>]oo`0087oo00<007ooOol0CWoo00=JeWooOol02Goo00=JeWooOol0 27oo00=JeWooOol02Goo00=JeWooOol0jgoo000POol00`00Oomoo`1>Ool00e[FOomoo`09Ool00e[F Oomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`3[Ool0021oo`03001oogoo04ioo`03F]Ioogoo00Uo o`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo0>]oo`0087oo00<007ooOol0CWoo00=JeWoo Ool02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol0jgoo000POol00`00Oomoo`1>Ool0 0e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`3[Ool0021oo`8004moo`03 F]Ioogoo00Uoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo0>]oo`0087oo00<007ooOol0 CWoo00=JeWooOol02Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol0jgoo000POol00`00 Oomoo`1>Ool00e[FOomoo`09Ool00e[FOomoo`08Ool00e[FOomoo`09Ool00e[FOomoo`3[Ool0021o o`03001oogoo04eoo`<0O`Yoo`03F]Ioogoo00Qoo`03F]Ioogoo00Uoo`03F]Ioogoo0>]oo`0087oo 00<007ooOol0C7oo1@1o2Goo00=JeWooOol027oo00=JeWooOol02Goo00=JeWooOol0jgoo000POol0 0`00Oomoo`1"], ImageRangeCache->{{{82.125, 467.375}, {350.25, 113.125}} -> {-9.85556, \ 0.0461635, 0.0861758, 0.000470952}}] }, Open ]], Cell[TextData[{ StyleBox["Fig. ", FontWeight->"Bold"], StyleBox[ CounterBox["Figure"], FontWeight->"Bold"], StyleBox[