(*********************************************************************** 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). 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Smith", "Subtitle", TextAlignment->Center], Cell[TextData[{ StyleBox["Abstract", FontWeight->"Bold"], "\nThe ", StyleBox["mathStatica", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " pacakage provides a completely general toolset for doing mathematical \ statistics with ", StyleBox["Mathematica", FontSlant->"Italic"], ". Common statistical operators are defined for taking expectations, \ finding probabilities, deriving transformations of random variables and so \ on... all from first principles. By taking full advantage of the latest \ Version 4 Assumptions technology, these functions automatically yield \ 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\tMurray.Smith@econ.usyd.edu.au", FontSize->9] }], "Text", CellMargins->{{53, 56.1875}, {10, Inherited}}, 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["mathStatica", 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"], " (600 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"] }, Closed]], 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["mathStatica", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " package can replace many reference works and extends the analysis to \ problems of arbitrary high order. This paper concentrates on just 4 \ functions, and illustrates how Assumptions technology is used to \ automatically generate crisp textbook quality answers." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Chapter"], ".3\tDesign Philosophy" }], "Subsubsection"], Cell[TextData[{ "The ", StyleBox["mathStatica", 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. 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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. 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"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[ \(TraditionalForm\`Y = \@X\)]], ". 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" }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["2.2\tIntroducing Assumptions Technology", "Subsubsection"], Cell[TextData[{ "The ", StyleBox["mathStatica", 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["mathStatica", 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. 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This means we can work with two or more pdf's, say:\n\n\t", Cell[BoxData[ \(TraditionalForm\`f(x; \ \ \[Theta])\)]], " \tassuming parameter ", Cell[BoxData[ \(TraditionalForm\`\[Theta] > 0\)]], ", and simultaneosuly \n\t\n\t", Cell[BoxData[ \(TraditionalForm\`h(x; \ \ \[Theta])\)]], " \tassuming parameter ", Cell[BoxData[ \(TraditionalForm\`\[Theta] < 0\)]], " \n\nand the package will automatically assume that \[Theta] is positive \ when working with ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], ", but negative when working with ", Cell[BoxData[ \(TraditionalForm\`h(x)\)]], ". " }], "Text"], Cell[TextData[{ "3. Finally, 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["mathStatica", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " package functions. 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\(TraditionalForm\`P(X \[LessEqual] x)\)]], "," }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Prob[x, \ f]\)], "Input"], Cell[BoxData[ \(1 - \((\((1 - p)\)\^\(1 + Floor[x]\)\ p\^r\ \[CapitalGamma][ 1 + r + Floor[x]]\ Hypergeometric2F1[1, 1 + r + Floor[x], 2 + Floor[x], 1 - p])\)/\((\[CapitalGamma][r]\ \[CapitalGamma][ 2 + Floor[x]])\)\)], "Output"] }, Open ]], Cell[TextData[{ "The mean ", Cell[BoxData[ \(TraditionalForm\`E[X]\)]], " and variance of ", Cell[BoxData[ \(TraditionalForm\`X\)]], " are given by:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Expect[x, \ f]\)], "Input"], Cell[BoxData[ \(\((\(-1\) + 1\/p)\)\ r\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Var[x, \ f]\)], "Input"], Cell[BoxData[ \(\(r - p\ r\)\/p\^2\)], "Output"] }, Open ]], Cell[TextData[{ "The probability generating function (pgf) is ", Cell[BoxData[ \(TraditionalForm\`E\ [t\^X]\)]], " :" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Expect[t\^x, \ f]\)], "Input"], Cell[BoxData[ \(p\^r\ \((1 + \((\(-1\) + p)\)\ t)\)\^\(-r\)\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["2.4\tMultivariate pdf's", "Subsubsection"], Cell[TextData[{ "The ", StyleBox["mathStatica", FontFamily->"Helv", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.570001, 0.2, 1], FontVariations->{"CompatibilityType"->0}], " toolset extends naturally to a multivariate setting. To illustrate, let \ us suppose that ", Cell[BoxData[ \(TraditionalForm\`X\)]], " and ", Cell[BoxData[ \(TraditionalForm\`Y\)]], " have joint pdf ", Cell[BoxData[ \(TraditionalForm\`f(x, \ y)\)]], " with support ", Cell[BoxData[ \(TraditionalForm\`x > 0\)]], ", ", Cell[BoxData[ \(TraditionalForm\`y > 0\)]], ":" }], "Text"], Cell[BoxData[{ \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(f\ \ \ = \ \ \ \[ExponentialE]\^\(\(-2\)\ \((x + y)\)\)\ \((\[ExponentialE]\^\(x + y\) + \((\ \(-2\) + \[ExponentialE]\^x)\)\ \((\(-2\) + \[ExponentialE]\^y)\)\ \ \[Alpha])\);\)\), "\[IndentingNewLine]", \(\(domain[ f]\ = \ {{x, \ 0, \ \[Infinity]}, \ {y, \ 0, \ \[Infinity]}}\ \ \ \ \ \ \ && \ \ \ \ {\(-1\)\ < \ \ \[Alpha] < 1};\)\)}], "Input"], Cell[TextData[{ "where parameter \[Alpha] is such that ", Cell[BoxData[ \(TraditionalForm\`\(-1\) < \[Alpha] < 1\)]], ". This is known as Morgenstern's (1956) Bivariate Exponential pdf. 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m .82633 .21745 L s .84407 .23761 m .84035 .23888 L s .8578 .2586 m .85407 .25986 L s .88439 .29928 m .88065 .3005 L s .89727 .31899 m .89353 .3202 L s .9099 .3383 m .90615 .3395 L s .93439 .37576 m .93063 .37693 L s .94627 .39394 m .94251 .3951 L s .95792 .41176 m .95416 .41291 L s gsave .91861 .21225 -61 -11.3715 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 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"\[IndentingNewLine]", RowBox[{"AxesLabel", " ", "\[Rule]", " ", RowBox[{"{", RowBox[{"\"\<\!\(\* StyleBox[\\\"x\\\",\\nFontSlant->\\\"Italic\\\"]\)\>\"", ",", " ", "\"\<\!\(\* StyleBox[\\\"y\\\",\\nFontSlant->\\\"Italic\\\"]\)\>\"", ",", " ", "\"\<\!\(\* StyleBox[\\\"f\\\",\\nFontSlant->\\\"Italic\\\"]\)\>\""}], "}"}]}]}], "]"}], ";"}]], "Input", Background->RGBColor[0.949996, 0.900008, 0.8]] }, Closed]], Cell[TextData[{ "Here is the cdf, namely ", Cell[BoxData[ \(TraditionalForm\`P(X \[LessEqual] x, \ Y \[LessEqual] y)\)]], "," }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Prob[{x, \ y}, \ f]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(\(-2\)\ \((x + y)\)\)\ \((\(-1\) + \ \[ExponentialE]\^x)\)\ \((\(-1\) + \[ExponentialE]\^y)\)\ \ \((\[ExponentialE]\^\(x + y\) + \[Alpha])\)\)], "Output"] }, Open ]], Cell[TextData[{ "Here is ", Cell[BoxData[ \(TraditionalForm\`Cov(X, \ Y)\)]], ", the covariance between ", Cell[BoxData[ \(TraditionalForm\`X\)]], " and ", Cell[BoxData[ \(TraditionalForm\`Y\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Cov[{x, \ y}, \ f]\)], "Input"], Cell[BoxData[ \(\[Alpha]\/4\)], "Output"] }, Open ]], Cell["More generally, here is the variance-covariance matrix:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Varcov[f]\ // \ MatrixForm\)\(\ \)\)\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", \(\[Alpha]\/4\)}, {\(\[Alpha]\/4\), "1"} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[TextData[{ "Here is the marginal density of ", Cell[BoxData[ \(TraditionalForm\`X\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Marginal[x, \ f]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(-x\)\)], "Output"] }, Open ]], Cell[TextData[{ "Here is the conditional pdf of ", Cell[BoxData[ \(TraditionalForm\`Y\)]], ", given ", Cell[BoxData[ \(TraditionalForm\`X = x\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Conditional[x, \ f]\)\(\ \)\)\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Here is the conditional pdf \"\>", "\[InvisibleSpace]", TagBox["f", HoldForm], "\[InvisibleSpace]", "\<\"(\"\>", "\[InvisibleSpace]", "y", "\[InvisibleSpace]", "\<\" | \"\>", "\[InvisibleSpace]", "x", "\[InvisibleSpace]", "\<\"):\"\>"}], SequenceForm[ "Here is the conditional pdf ", HoldForm[ f], "(", y, " | ", x, "):"], Editable->False]], "Print"], Cell[BoxData[ \(\[ExponentialE]\^\(x - 2\ \((x + y)\)\)\ \((\[ExponentialE]\^\(x + y\) \ + \((\(-2\) + \[ExponentialE]\^x)\)\ \((\(-2\) + \[ExponentialE]\^y)\)\ \ \[Alpha])\)\)], "Output"] }, Open ]], Cell[TextData[{ "Here is the bivariate mgf ", Cell[BoxData[ \(TraditionalForm\`E[\[ExponentialE]\^\(\(t\_1\) X\ + \ \(t\_2\) \ Y\)]\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(mgf\ = \ Expect[\ \[ExponentialE]\^\(\(t\_1\) x\ + \ \(t\_2\) y\), \ f]\)\(\ \)\)\)], "Input"], Cell[BoxData[ \(\(4 - 2\ t\_2 + t\_1\ \((\(-2\) + \((1 + \[Alpha])\)\ \ t\_2)\)\)\/\(\((\(-2\) + t\_1)\)\ \((\(-1\) + t\_1)\)\ \((\(-2\) + t\_2)\)\ \ \((\(-1\) + t\_2)\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "Differentiating the mgf is one way to derive moments. Here is the product \ moment ", Cell[BoxData[ \(TraditionalForm\`E[X\^2\ Y\^2]\)]], ": " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[mgf, \ {t\_1, 2}, \ {t\_2, \ 2}] /. \ t\__ \[Rule] \ 0\ // \ Simplify\)], "Input"], Cell[BoxData[ \(4 + \(9\ \[Alpha]\)\/4\)], "Output"] }, Open ]], Cell["which we could otherwise have found directly with: ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Expect[\(x\^2\) y\^2, \ f]\)], "Input"], Cell[BoxData[ \(4 + \(9\ \[Alpha]\)\/4\)], "Output"] }, Open ]], Cell[TextData[{ "Multivariate transformations pose no problem either. For instance, let ", Cell[BoxData[ \(TraditionalForm\`U = Y\/\(1 + X\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`V = 1\/\(1 + X\)\)]], ". Then our transformation equation is:" }], "IndentText"], Cell[BoxData[ \(\(\(eqn\ = \ {u \[Equal] \ y\/\(1 + x\), \ v\ \[Equal] 1\/\(1 + x\)};\)\(\ \ \ \)\)\)], "Input"], Cell[TextData[{ "Using ", StyleBox["Transform", FontFamily->"Courier"], ", we can find the joint density of random variables ", Cell[BoxData[ \(TraditionalForm\`U\)]], " and ", Cell[BoxData[ \(TraditionalForm\`V\)]], ", denoted ", Cell[BoxData[ \(TraditionalForm\`g(u, \ v)\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(g\ = \ Transform[eqn, \ f]\)], "Input"], Cell[BoxData[ \(\(\[ExponentialE]\^\(\(\(-2\) - 2\ u + v\)\/v\)\ \((4\ \[ExponentialE]\ \ \[Alpha] - 2\ \[ExponentialE]\^\(1\/v\)\ \[Alpha] - 2\ \[ExponentialE]\^\(\(u \ + v\)\/v\)\ \[Alpha] + \[ExponentialE]\^\(\(1 + u\)\/v\)\ \((1 + \[Alpha])\))\ \)\)\/v\^3\)], "Output"] }, Open ]], Cell["while the extremities of the new random variables are:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TransformExtremum[eqn, \ f]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"u", ",", "0", ",", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]}], "}"}], ",", \({v, 0, 1}\)}], "}"}]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["2.5\tBipartite pdf's (kinks)", "Subsubsection"], Cell[TextData[{ "Some density functions take a bipartite form. To illustrate, let us \ suppose ", Cell[BoxData[ \(TraditionalForm\`X\)]], " is a continuous random variable ", Cell[BoxData[ \(TraditionalForm\`x\ \[Element] \ \((0, 1)\)\)]], " with pdf:" }], "Text"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{\(f(x)\), " ", "=", " ", RowBox[{ StyleBox["{", AutoStyleOptions->{"UnmatchedBracketStyle"->None}, SpanMaxSize->Infinity], StyleBox[ RowBox[{ StyleBox[" ", AutoStyleOptions->{"UnmatchedBracketStyle"->None}, SpanMaxSize->Infinity], " "}]], GridBox[{ {\(2 \((\(\[Alpha] - x\)\/\[Alpha])\)\), \(\(\ \ \ \ \ \ \)\(if\ \ x < \[Alpha]\)\)}, {\(2 \((\(x - \[Alpha]\)\/\(1 - \[Alpha]\))\)\), \(\(\ \ \ \ \ \ \)\(otherwise\)\)} }, ColumnAlignments->{Left}]}]}], TraditionalForm]]]], "UnNumberedEquation"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`0 < \[Alpha] < 1\)]], ". We enter this as:" }], "Text"], Cell[BoxData[{ \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(f\ \ \ \ = \ \ \ \ If[ x < \[Alpha]\ , 2 \( \[Alpha] - x\)\/\[Alpha], 2 \( x - \[Alpha]\)\/\(1 - \[Alpha]\)]\ ;\)\ \ \ \), "\ \[IndentingNewLine]", \(\(domain[ f]\ \ = \ \ \ {x, 0, 1}\ \ && \ \ {0 < \[Alpha] < 1};\)\)}], "Input", GeneratedCell->False], Cell[TextData[{ "This is known as the Inverse Triangular distribution, as is clear from a \ plot of ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(PlotDensity[ f /. \ \[Alpha] \[Rule] {1\/4, 1\/2, \ 3\/4}];\)\)], "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 -5.55112e-17 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Cell[StyleData["Graphics", "Printout"], CellMargins->{{55, 30}, {15, 15}}, ImageSize->{0.0625, 0.0625}, ImageMargins->{{35, Inherited}, {Inherited, 0}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], CellMargins->{{9, Inherited}, {Inherited, Inherited}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["CellLabel", "Printout"], CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontFamily->"Courier", FontSize->8, FontSlant->"Italic", FontColor->GrayLevel[0]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Automatic Numbering", "Section"], Cell[CellGroupData[{ Cell[StyleData["Example"], CellMargins->{{18, 10}, {Inherited, 25}}, TextJustification->1, LineSpacing->{1, 2}, CounterIncrements->"Example", Background->RGBColor[0.8, 0.850004, 1]], Cell[StyleData["Example", "Printout"], CellMargins->{{18, 30}, {Inherited, 20}}, PageBreakBelow->False, LineSpacing->{1, 3}, FontSize->10, Background->None] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["FigureLabel"], ShowCellBracket->False, ShowGroupOpenCloseIcon->True, CellMargins->{{55, Inherited}, {15, 5}}, CellHorizontalScrolling->True, PageBreakAbove->False, ShowCellLabel->False, TextAlignment->Left, CounterIncrements->"Figure", FontColor->RGBColor[0.700008, 0.100008, 0]], Cell[StyleData["FigureLabel", "Printout"], CellMargins->{{18, 30}, {15, 1}}, TextAlignment->Center, FontSize->9, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["TableLabel"], ShowCellBracket->False, ShowGroupOpenCloseIcon->True, CellMargins->{{55, Inherited}, {15, 5}}, CellHorizontalScrolling->True, PageBreakAbove->False, ShowCellLabel->False, TextAlignment->Left, CounterIncrements->"Table", FontColor->RGBColor[0.700008, 0.100008, 0]], Cell[StyleData["TableLabel", "Printout"], CellMargins->{{18, 30}, {15, 1}}, TextAlignment->Center, FontSize->9, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ 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below define styles useful for making hypertext \ ButtonBoxes. 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If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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