\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Arial;}{\f4\fmodern\fprq1 Courier New;}{\f5\froman\fprq2 Times New Roman;}{\f6\fswiss\fprq2 Verdana;}{\f7\fswiss\fprq2 Helvetica;}} {\colortbl\red0\green0\blue0;\red0\green0\blue255;\red255\green0\blue0;\red0\green128\blue0;} \deflang1031\pard\ri4\plain\f4\fs20\cf0\b _________________________________________________________________________________ \par \plain\f4\fs20\cf0 \par Inhalt....: Grafische Interpretation der N\'e4herung von DeMoivre-Laplace \par Kategorie.: Arbeitsblatt \par Mathematik: Stochastik, Statistik \par MuPAD.....: 3.0.0 \par Datum.....: 2002-01-17 \par Autoren...: Julia Faflek \par Funktionen: stats::binomialPF, stats::normalPDF, plot, plot::Polygon2d \par Funktionen: plot::Function2d \par \plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \plain\f3\fs36\cf0\b \par \plain\f3\fs40\cf0\b Die N\'e4herung von DeMoivre-Laplace\plain\f3\fs36\cf0\b \par \plain\f3\fs22\cf0 \par \plain\f3\fs24\cf3 Das Beispiel liefert eine grafische Interpretation der Approximation von standardisierten \par binomialverteilten Zufallsvariablen durch die Normalverteilung und veranschaulicht die wachsende \par Angleichung an die Gau\'df-Glocke mit Zunahme des Stichprobenumfangs. \par \plain\f3\fs28\cf0 \par Wir wollen\plain\f3\fs28 eine Prozedur schreiben, die bei Eingabe von\plain\f3\fs28\cf2 n \plain\f3\fs28 und \plain\f3\fs28\cf2 p\plain\f3\fs28 eine Glocke \par einer binomialverteilten Zufallsvariable zeichnet. Dabei bezeichne \plain\f3\fs28\cf2 n\plain\f3\fs28 die Anzahl \par der Durchf\'fchrungen eines Bernoulli-Experiments und \plain\f3\fs28\cf2 p\plain\f3\fs28 die Erfolgswahrschein- \par lichkeit.\plain\f4\fs22\cf2 \par \plain\f3\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}Glocke := proc(n, p) \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 local Punkte; \par begin \par Punkte := [0 $ n+1]; \par for j from 0 to n do \par Punkte[j+1] := [j, stats::binomialPF(n, p)(j)] \par end_for; \par plot::Polygon2d(Punkte, args(3..args(0))); \par end_proc: \par \pard\ri4\plain\f3\fs28\cf0 \par Wir erhalten mit dem Aufruf \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}plot(Glocke(10, 0.5))\plain\f3\fs28\cf2 \par \pard\ri4\plain\f3\fs28\cf0 \par die gew\'fcnschte Glocke. \plain\f3\fs28 Um einen Zusammenhang zwischen den Parametern \par und der Glocke zu erkennen, zeichnen wir mehrere Glocken mit unter- \par schiedlichen Parametern in eine Graphik.\plain\f3\fs28\cf0 \par \plain\f3\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}plot(plot::Scene2d(Glocke(4, 0.5, Color = RGB::Blue), \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 Glocke(16, 0.5, Color = RGB::Red), \par Glocke(64, 0.5, Color = RGB::Black))) \par \pard\ri4\plain\f3\fs28\cf0 \par Wie man gut erkennen kann, halbiert sich \plain\f3\fs28 die Glockenh\'f6he bei jedem \par \'dcbergang und die Fl\'e4che wird ungef\'e4hr verdoppelt. \par \par Im Vergleich zu den Parametern k\'f6nnen wir folgenden Zusammenhang \par feststellen: \par \plain\f3\fs28\i \par Die Gipfelh\'f6he wird durch den Faktor 1/Standardabweichung verkleinert \par und die Fl\'e4che wird mit der Standardabweichung multipliziert.\plain\f3\fs28 \par \par Um dieses Verhalten auszugleichen, standardisieren wir die binomial- \par verteilte Zufallsvariable, das hei\'dft wir setzen f\'fcr die Werte auf der x-Achse \par {\pict\wmetafile8\picw2264\pich1347\picscalex99\picscaley99\picwgoal1292\pichgoal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}\plain\f3\fs28 \par und f\'fcr die y-Werte \par 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}\plain\f3\fs28 \par wobei sigma die Standardabweichung und mue den Erwartungswert bezeichne. \par \par Durch Verwendung dieser Werte ver\'e4ndern wir die Prozedur \plain\f3\fs28\cf2 Glocke\plain\f3\fs28 wie folgt.\plain\f3\fs28\cf2 \par \plain\f3\fs28\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}Glocke := proc(n, p) \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 local Punkte; \par begin \par mue := n*p; \par sigma := sqrt(n*p*(1-p)); \par Punkte := [0 $ n+1]; \par for j from 0 to n do \par Punkte[j+1] := [(j-mue)/sigma, \par sigma*stats::binomialPF(n,p)(j)] \par end_for; \par plot::Polygon2d(Punkte, args(3..args(0))); \par end_proc: \par \pard\ri4\plain\f3\fs28\cf0 \par Damit \plain\f3\fs28 k\'f6nnen wir nun Glocken zeichnen lassen, deren Gipfel sich auf der \par y-Achse befindet und die alle ungef\'e4hr die gleiche Fl\'e4che unter der Glocke \par beschreiben.\plain\f6\fs24 \par \plain\f3\fs28\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}plot( plot::Scene2d( Glocke(4, 0.5, Color = RGB::Blue), \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 Glocke(16, 0.5, Color = RGB::Red), \par Glocke(64, 0.5, Color = RGB::Black))) \par \pard\ri4\plain\f3\fs28 \par Die so erzeugten Glocken n\'e4hern sich der sogenannten \plain\f3\fs28\i Gau\'df-Glocke\plain\f3\fs28 an. \par Zum direkten Vergleich zeichnen wir die Gau\'df-Funktion, die von der \par MuPAD Bibliothek \plain\f3\fs28\cf2 stats\plain\f3\fs28 zur Statistik bereitgestellt wird: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}f := stats::normalPDF(0,1): \par {\pntext\f1\'b7\tab}plot(plot::Function2d(f(x), x = -7.5..7.5)) \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}plot(plot::Scene2d(Glocke(4, 0.5, Color = RGB::Blue), \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 Glocke(16, 0.5, Color = RGB::Blue), \par Glocke(64, 0.5, Color = RGB::Black))) \par \pard\ri4\plain\f3\fs28 \par Diese Graphiken zeigen deutlich, dass die Binomialverteilung f\'fcr gro\'dfe Werte \plain\f3\fs28\cf2 \par n\plain\f3\fs28 durch die Normalverteilung approximiert werden kann.\plain\f6\fs24 \par \plain\f4\fs20\cf0\b _______________________________________________________________________________ \par \plain\f3\fs22\cf0 \par \plain\f3\fs22\cf3\b Anmerkungen:\plain\f3\fs22\cf3 \par \plain\f3\fs20\cf3\b 1\plain\f3\fs20\cf3 . Weitere Anregungen finden Sie in der Buchreihe \plain\f3\fs20\cf2 Mathematik 1 x anders\plain\f3\fs20\cf3 . In dieser Reihe \par wird eine Vielzahl unterschiedlichster mathematischer Probleme mit MuPAD gel\'f6st. Die \par B\'fccher k\'f6nnen unter \plain\f7\fs20\cf1 www.schule.mupad.de/literatur\plain\f3\fs20\cf3 kostenfrei kopiert werden. \plain\f3\fs20\cf1 \par \par \plain\f4\fs20\cf0\b _______________________________________________________________________________ \par \par \par }