\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\froman Times New Roman;}{\f4\fswiss\fprq2 Arial;}{\f5\fmodern\fprq1 Courier New;}{\f6\fswiss\fprq2\fcharset1 Arial;}{\f7\fswiss\fprq2 Helvetica;}{\f8\froman\fprq2 Times New Roman;}} {\colortbl\red0\green0\blue0;\red255\green0\blue0;\red255\green255\blue255;\red0\green128\blue0;\red51\green102\blue255;\red0\green0\blue255;} \deflang1031\pard\ri4\plain\f5\fs20\cf0\b _____________________________________________________________________________________ \par \par Inhalt....: Absorbierende Markoff-Kette \par Kategorie.: Unterrichtsmaterial \par Mathematik: Lineare Algebra, Stochastik \par MuPAD.....: 3.1.0 \par Datum.....: 2005-01-12 \par Autoren...: Monika v. zur M\'fchlen \par Funktionen: matrix, linalg::transpose \par _____________________________________________________________________________________ \par \plain\f6\fs24\cf3\b \par Markoff-Ketten sind ein interessantes Thema f\'fcr einen anwendungsorientierten Mathe- \par matikunterricht, das Inhalte der Stochastik mit der Linearen Algebra verbindet. 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}\plain\f4\fs24 \par \par Eine Ameise startet in Zelle 4 und wechselt von Minute zu Minute in eine der Nachbarzellen, dabei \par hat sie einen \'84Linksdrall" und geht jeweils mit einer Wahrscheinlichkeit von 0,6 nach links und mit \par einer Wahrscheinlichkeit von 0,4 nach rechts. Wenn sie in einer der Randzellen gelandet ist, bleibt \par sie dort f\'fcr immer gefangen. \par \par a)\plain\f4\fs24\cf2 -------\plain\f4\fs24 Stelle die \'dcbergangsmatrix U auf, die diesen Prozess beschreibt. \par b)\plain\f4\fs24\cf2 -------\plain\f4\fs24 Bestimme die Wahrscheinlichkeiten, mit denen die Ameise in Zelle 1 bzw. in Zelle 6 gefangen \par \plain\f4\fs24\cf2 ----------\plain\f4\fs24 wird. \plain\f4\fs24\i \par \plain\f4\fs24 c)\plain\f4\fs24\cf2 -------\plain\f4\fs24 Wie lange braucht die Ameise bis zu ihrer Gefangennahme? Berechne die mittlere Zeitdauer. \plain\f4\fs24\i \par \plain\f4\fs24\cf0 \par \par \plain\f4\fs24\cf0\b L\'f6sung zu a): \par \plain\f4\fs22\cf0\b \par \plain\f4\fs24\cf0 Es gilt z.B. f\'fcr die Wahrscheinlichkeit, dass die Ameise sich in Zelle 3 aufh\'e4lt: \par z3:= 0.6*z4 + 0.4*z2 \par oder in Zelle z1: \par z1:= 0.6*z2 + 1*z1 (Wenn die Ameise bereits in z1 gefangen ist, bleibt sie auch dort!) \par \par Aus diesen \'dcberlegungen ergibt sich die folgende \'dcbergangsmatrix: \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs24\cf1 {\pntext\f1\'b7\tab}U:= matrix([[1,0.6,0,0,0,0], [0,0,0.6,0,0,0],[0,0.4,0,0.6,0,0], \par \pard\li600\ri1\fi-300\plain\f5\fs24\cf1 [0,0,0.4,0,0.6,0],[0,0,0,0.4,0,0],[0,0,0,0,0.4,1]]) \par \par \pard\ri4\plain\f4\fs24\cf0 Zu Beginn befindet sich die Ameise mit Wahrscheinlichkeit 1 in Zelle 4 und mit Wahrscheinlichkeit 0 \par in allen anderen Zellen. \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs24\cf1 {\pntext\f1\'b7\tab}v0:= matrix([0,0,0,1,0,0]) \par \pard\li600\ri1\fi-300\plain\f5\fs24\cf1 \par \pard\ri4\plain\f4\fs24\cf0 Nach einer Minute: \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs24\cf1 {\pntext\f1\'b7\tab}v1:= U*v0 \par \pard\li600\ri1\fi-300\plain\f5\fs24\cf1 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs24\cf1 {\pntext\f1\'b7\tab}v2:= U*v1 \par \pard\li600\ri1\fi-300\plain\f5\fs24\cf1 \par \pard\ri4\plain\f4\fs24\cf0 Und nach 10 Minuten: \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs24\cf1 {\pntext\f1\'b7\tab}DIGITS:= 4: v10:= U^10*v0 \par \pard\li600\ri1\fi-300\plain\f5\fs24\cf1 \par \pard\ri4\plain\f4\fs24\cf0\b L\'f6sung zu b): \par \plain\f4\fs24\cf0 \par Mit welchen Wahrscheinlichkeiten landet die Ameise, von einer bestimmten Zelle aus gesehen, \par im Endzustand z1? \par \par F\'fcr die \'dcbergangswahrscheinlichkeiten in den Zustand z1 gilt: \par \par a1 = 1\plain\f4\fs24\cf2 -------------------------------------------------\plain\f4\fs24\cf0 (Wenn die Ameise in z1 ist, bleibt sie dort.) \par a2 = 0,4*a3 + 0,6*a1 = 0,4*a3 + 0,6\plain\f4\fs24\cf2 --------\plain\f4\fs12\cf2 -\plain\f4\fs24\cf0 (Von z2 aus kann sie direkt nach z1 gelangen oder den \par \plain\f4\fs24\cf2 -----------------------------------------------------------\plain\f4\fs24\cf0 Umweg \'fcber z3 nehmen) \par a3 = 0,4*a4 + 0,6*a2\plain\f4\fs24\cf2 -----------------------------\plain\f4\fs24\cf0 (Von z3 aus kann sie nur \'fcber z2 oder z4 nach z1 gelangen,) \par \par a4 = 0,4*a5 + 0,6*a3\tab \tab \tab \par a5 = 0,6*a4\tab \tab \tab \tab \tab \par a6 = 0\plain\f4\fs24\cf2 -------------------------------------------------\plain\f4\fs24\cf0 (Von z6 aus kann die Ameise nicht mehr nach z1 gelangen!) \par \par Offenbar lassen sich diese Gleichungen aus der \'dcbergangsmatrix ableiten, \par \par \plain\f3\fs24\cf0 1\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf3\b 0.6\plain\f3\fs24\cf2 ___\plain\f3\fs12\cf2 _\plain\f3\fs24\cf3\b 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf3\b 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf3\b 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf0 0\tab \par 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf1\b 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf1\b 0.6\plain\f3\fs24\cf2 ___\plain\f3\fs12\cf2 _\plain\f3\fs24\cf1\b 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf1\b 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf0 0\tab \par 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf1\b 0.4\plain\f3\fs24\cf2 ___\plain\f3\fs12\cf2 _\plain\f3\fs24\cf1\b 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf1\b 0.6\plain\f3\fs24\cf2 ___\plain\f3\fs12\cf2 _\plain\f3\fs24\cf1\b 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf0 0\tab \par 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf1\b 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf1\b 0.4\plain\f3\fs24\cf2 ___\plain\f3\fs12\cf2 _\plain\f3\fs24\cf1\b 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf1\b 0.6\plain\f3\fs24\cf2 ___\plain\f3\fs12\cf2 _\plain\f3\fs24\cf0 0\tab \par 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf1\b 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf1\b 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf1\b 0.4\plain\f3\fs24\cf2 ___\plain\f3\fs12\cf2 _\plain\f3\fs24\cf1\b 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf0 0\tab \par 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf4\b 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf4\b 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf4\b 0\plain\f3\fs24\cf2 _____\plain\f3\fs24\cf4\b 0.4\plain\f3\fs24\cf2 ___\plain\f3\fs12\cf2 _\plain\f3\fs24\cf0 1\tab \plain\f4\fs24\cf0 \par \par ......... wenn man die Zeilen wegl\'e4sst, die sich auf die Endzust\'e4nde beziehen (das sind hier die \par Zeilen 1 und 6), und die Matrix spaltenweise liest. \par Durch Transponierung der rot geschriebenen Teilmatrix ergibt sich zun\'e4chst die Matrix Q: \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs24\cf1 {\pntext\f1\'b7\tab}Q:= linalg::transpose( matrix( \par \pard\li600\ri1\fi-300\plain\f5\fs24\cf1 [[0,0.6,0,0], [0.4,0,0.6,0], [0,0.4,0,0.6], [0,0,0.4,0]] \par )) \par \par \pard\ri4\plain\f4\fs24\cf0 Die Matrix Q wird mit dem Vektor der Wahrscheinlichkeiten multipliziert, die Zeile der Matrix U, \par die zu z1 geh\'f6rt, wird ebenfalls transponiert und hinzuaddiert. Es ergibt sich das oben aufgef\'fchrte \par Gleichungssystem. \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs24\cf1 {\pntext\f1\'b7\tab} matrix([a2, a3, a4, a5]) = \par \pard\li600\ri1\fi-300\plain\f5\fs24\cf1 Q*matrix([a2, a3, a4, a5]) + matrix([0.6, 0, 0, 0]) \par \par \pard\ri4\plain\f4\fs24\cf0 Die Gleichung wird aufgel\'f6st mit Hilfe der inversen Matrix. Es gilt: \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs24\cf1 {\pntext\f1\'b7\tab}a:= matrix([a2, a3, a4, a5]): \par \pard\li600\ri1\fi-300\plain\f5\fs24\cf1 print(Typeset,hold((1 - Q)*a) = matrix([0.6, 0, 0, 0])) \par \par \pard\ri4\plain\f4\fs24\cf0 Daraus folgt: \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs24\cf1 {\pntext\f1\'b7\tab}hold(a) = (1-Q)^(-1)*matrix([0.6, 0, 0, 0]) \par \pard\li600\ri1\fi-300\plain\f5\fs24\cf1 \par \pard\ri4\plain\f4\fs24\cf0 Die Matrix F wird \plain\f4\fs24\cf0\b Fundamentalmatrix \plain\f4\fs24\cf0 der Markoff-Kette genannt. \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs24\cf1 {\pntext\f1\'b7\tab}F:= (1-Q)^(-1) \par \pard\li600\ri1\fi-300\plain\f5\fs24\cf1 \par \pard\ri4\plain\f4\fs24\cf0 Berechnung der \plain\f4\fs24\cf0\b Absorptionswahrscheinlichkeiten\plain\f4\fs24\cf0 mit Hife von F: \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs24\cf1 {\pntext\f1\'b7\tab}a:= F*matrix([[0.6], [0], [0], [0]]) // Absorption in z1 \par \pard\li600\ri1\fi-300\plain\f5\fs24\cf1 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs24\cf1 {\pntext\f1\'b7\tab}b:= F*matrix([[0], [0], [0], [0.4]]) // Absorption in z6 \par \pard\li600\ri1\fi-300\plain\f5\fs24\cf1 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs24\cf1 {\pntext\f1\'b7\tab}DIGITS:= 3: \par {\pntext\f1\'b7\tab}a[3] \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs24\cf1 {\pntext\f1\'b7\tab}b[3] \par \pard\ri4\plain\f4\fs24\cf0 Die Ameise wird ihre Wanderung mit einer Wahrscheinlichkeit von 64% im ersten Feld beenden und mit \par einer Wahrscheinlichkeit von 36% im zweiten Feld.. \par \plain\f4\fs24\cf0\b \par L\'f6sung zu c): \par \plain\f4\fs24\cf0 Mit der Fundamentalmatrix kann auch die mittlere Schrittl\'e4nge bestimmt werden: \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs24\cf1 {\pntext\f1\'b7\tab}m:= F*matrix([[1], [1], [1], [1]]) \par \pard\li600\ri1\fi-300\plain\f5\fs24\cf1 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs24\cf1 {\pntext\f1\'b7\tab}m[3] \par \pard\ri4\plain\f4\fs24\cf0 Die Ameise ben\'f6tigt im Mittel 6 Minuten bis zur Gefangennahme, wenn sie in Feld 4 startet. \par \par \plain\f4\fs24\cf0\b Variation: \par \plain\f4\fs22 Die Ameise startet in Zelle 3, und es gelten nun die folgenden \'dcbergangswahrscheinlichkeiten: \par - mit einer Wahrscheinlichkeit von 0,1 bleibt sie in der jeweiligen Zelle, \par - mit einer Wahrscheinlichkeit von 0,5 geht sie nach links, \par - mit einer Wahrscheinlichkeit von 0,4 nach rechts. \par Wenn sie in einer der Randzellen gelandet ist, bleibt sie dort f\'fcr immer gefangen. \par \plain\f4\fs24\cf0\b \par \plain\f5\fs20\cf0\b ________________________________________________________________________________ \par \plain\f4\fs22\cf0 \par \plain\f4\fs22\cf3\b Anmerkungen:\plain\f4\fs20\cf3 \par \pard\plain\f4\fs20\cf3\b 1\plain\f4\fs20\cf3 . Selbstlernmaterial zur Matrizenrechnung unter: \par \pard\ri4\plain\f4\fs20\cf3 http://www.learnline.nrw.de/angebote/selma/foyer/projekte/hammproj3/index.html \par http://www.learnline.nrw.de/angebote/selma/foyer/projekte/dinslakenproj3/index.html \par \par \plain\f4\fs20\cf3\b 2.\plain\f4\fs20\cf3 Weitere Anregungen finden Sie in der Buchreihe \plain\f4\fs20\cf1 Mathematik 1 x anders\plain\f4\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\cf5 www.schule.mupad.de/literatur\plain\f4\fs20\cf3 kostenfrei kopiert werden. \par \plain\f5\fs20\cf0\b _______________________________________________________________________________ \par \plain\f5\fs22\cf1 \par }