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0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 271 "" 0 "" {TEXT 354 136 "Ellen Haller \+ Isolde-Kurz-Gymnasium \+ 09. Juni 2002" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 355 7 "Quelle:" } {TEXT 356 46 " Lambacher Schweizer - Analysis Leistungskurs" }{TEXT 357 0 "" }{TEXT 358 16 " (Seite 162/163)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 277 0 "" }{TEXT 260 0 "" }{TEXT 261 22 "Glockenkurve von GAUSS" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 261 "" 0 "" {TEXT 266 64 "Was f\374r besondere Eigen schaften hat eigentlich die Glockenkurve?" }}}{EXCHG {PARA 262 "" 0 " " {TEXT 278 82 "Um dies heraus zu bekommen, f\374hre ich als erstes ei ne Funktionsuntersuchung durch." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 259 40 "Funktionsuntersuchung einer Glockenkur ve" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 348 28 "Funktion einer Glockenkurve:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f:=x->exp(-1/2*((x-mu)/sigma )^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operator G%&arrowGF(-%$expG6#,$*&*$),&9$\"\"\"%#muG!\"\"\"\"#\"\"\"F9*$)%&sigma G\"\"#F9!\"\"#F7F8F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 279 0 "mu;" "6#%#muG" }{TEXT 280 28 " ist der Mittelwert und die " }{XPPEDIT 281 0 "sigma;" "6#%&sigmaG " }{TEXT 282 19 " Standardabeichung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "mu:=2;sigma:=0.5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#muG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaG$\"\"&!\"\"" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#, $*$),&%\"xG\"\"\"!\"#F+\"\"#\"\"\"$!+++++?!\"*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 58 "p1:=plot(f(x),x=-0.5..4,-0.5..1.2,colour=red,t hickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 34 "Ist es wirklich eine Glockenkurve?" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 3 "p1;" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6#7`o7$$!1+++++++]!#;$\"1r'y?<`ms$!#@7$$!1++] i!G\">SF*$\"1T?xx>w[(*F-7$$!1+]PMmnlJF*$\"1P*=/*=P\"=#!#?7$$!1++vV7)e? #F*$\"1w&[R$)z;@&F87$$!1++D1PsR7F*$\"1'p*)p0?m?\"!#>7$$!10+vo9e\"y#!#< $\"1V#Q[D07o#FC7$$\"1++Dc@OLhFG$\"1)fk*\\WSQaFC7$$\"1**\\i!*pUO:F*$\"1 -utf$HQ4\"!#=7$$\"1+]i!z)3\"\\#F*$\"1qQw,\\)Q<#FT7$$\"1**\\7y*)oUMF*$ \"1N!odr$HdTFT7$$\"1,+]Pn_@WF*$\"1[ec\"yo))z(FT7$$\"1++vovo$G&F*$\"1IG _&[;\\J\"FG7$$\"1,+]ikFaiF*$\"1j!ez([y%G#FG7$$\"1****\\(o])GsF*$\"1+l2 Jq*4$QFG7$$\"1,+]PN.o\")F*$\"1G^#p/i:3'FG7$$\"1**\\iS:!4-*F*$\"1!f\"[! ))QU(*)FG7$$\"1++v3W].5!#:$\"19%>_^BCP\"F*7$$\"1+++&e:%*3\"Fjp$\"1'=3A nrX!>F*7$$\"1+Dc12N*=\"Fjp$\"1(>\")f(4h'o#F*7$$\"1]7`k/eL7Fjp$\"1&HI^E 9))3$F*7$$\"1++]A-\"yF\"Fjp$\"1BWb'y\\N_$F*7$$\"1]7.xsLE8Fjp$\"1p%QS@T Z.%F*7$$\"1+DcJV'[P\"Fjp$\"1\")[7u3ywXF*7$$\"1+]7jN2@9Fjp$\"1&3VZM\"\\ :^F*7$$\"1+vo%z#Gn9Fjp$\"1jR^]O)*ocF*7$$\"1]i:!G(\\::Fjp$\"1RSy^uB`iF* 7$$\"1+]ilFjp$\"1&>+z#)p:\"**F*7$ $\"1vVtirnc>Fjp$\"1U6/+I`i**F*7$$\"1]7.ik**z>Fjp$\"1)[&yq.+#***F*7$$\" 1(oz;6c;*>Fjp$\"1n'*[(o2')***F*7$$\"1D\"G8w:L+#Fjp$\"1gP&p6!y****F*7$$ \"1il(4Tv\\,#Fjp$\"1h&[qu:b***F*7$$\"1+]ig]jE?Fjp$\"1P!p=`@e)**F*7$$\" 1+Dc'H<[2#Fjp$\"1UTTzq*F*7$$\"1+ ]7oLFeA#Q'oE)F*7$$\"1](=UQF\"fBFjp$\"1eGh)fxjs(F*7$$\"1+]( ohm(4CFjp$\"18ajgj`ZrF*7$$\"1+vo>#o_X#Fjp$\"12c]xAX1mF*7$$\"1++]A)p2]# Fjp$\"1\"pEb=of0'F*7$$\"1+]i&\\_$\\DFjp$\"16I&o5C&oaF*7$$\"1++vo^$zf#F jp$\"1OGP&=i;*[F*7$$\"1]iS\"He>k#Fjp$\"1+=Bibw&Q%F*7$$\"1+D199)fo#Fjp$ \"1xWMGQ#=!RF*7$$\"1]i::.6MFFjp$\"1YrMoBJ.MF*7$$\"1++D;#RAy#Fjp$\"1y/* )\\[6THF*7$$\"1+Dc1.\"G(GFjp$\"10Ys\\)o#z@F*7$$\"1+D\"G>$[nHFjp$\"1cW7 JU3Q:F*7$$\"1++vVK/gIFjp$\"1eO>yPvc5F*7$$\"1+D1R]%p:$Fjp$\"1j,-(4:m(oF G7$$\"1,++&)HF]KFjp$\"1/mQP(*p(Q%FG7$$\"1+]P*G9dM$Fjp$\"1Z([d3+Kn#FG7$ $\"1+Dc\"Hl.W$Fjp$\"1CX_(oQvd\"FG7$$\"1++]K(Rt_$Fjp$\"1'p;\"=hA9%*FT7$ $\"1+](oDAqi$Fjp$\"1pQtZ4h>]FT7$$\"1+++&\\zhr$Fjp$\"1BDHW:ZlFFT7$$\"1+ Dc1(R7\"QFjp$\"1$oDA+=UT\"FT7$$\"1+vVeWA-RFjp$\"1qFz@_F&>(FC7$$\"\"%\" \"!$\"1>^-ziiaLFC-%'COLOURG6&%$RGBG$\"*++++\"!\")Fj`lFj`l-%+AXESLABELS G6$Q\"x6\"%!G-%*THICKNESSG6#\"\"$-%%VIEWG6$;$!\"&!\"\"Fh`l;Fbbl$\"#7Fd bl" 1 2 0 1 3 2 6 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 283 129 "Nach dem Schaubild muss die Glockenkurve symmetrisch zu x=2 s ein, einen Hochpunkt, zwei Wendepunkte und keine Nullstellen haben. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" } {TEXT 347 9 "Symmetrie" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 268 64 "Wenn die Funktion symmetrisch i st muss gelten: f(mu-x) = f(mu+x)" }}{PARA 0 "" 0 "" {TEXT 284 51 "Die s stelle ich anhand einer kleinen Prozedur fest:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "Symmetrie:=proc(f) \nif evalb(f(mu-x)=f(mu+x ))=true then print(`Die Funktion ist symmetrisch zu mu`); else print(` Die Funktion ist nicht symmetrisch`) fi;end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Symmetrie(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%CDie~Funktion~ist~symmetrisch~zu~muG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 285 32 "Die Funktion ist symmetrisch zu " }{XPPEDIT 286 0 "mu;" "6#%#muG" }{TEXT 287 45 ". In Diesem Beispiel also symmetrisch zu x=2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 346 11 "Nullstellen" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "xnst:=solve(f(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xnstG6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 288 38 "Die Glocken kurve hat keine Nullstellen" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 345 11 "Ableitungen" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 289 13 "1. \+ Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "fs:=unapply(( diff(f(x),x),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fsGR6#%\"xG6\" 6$%)operatorG%&arrowGF(*&,&9$$!+++++S!\"*$\"+++++!)F1\"\"\"F4-%$expG6# ,$*$),&F.F4!\"#F4\"\"#\"\"\"$!+++++?F1F4F(F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "p2:=plot(fs(x),x=-1..5,-5..2.1,colour=green): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 290 13 "2. 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"6#%KDie~2.~Ableitung~der~Glockenkurv e~ist~blauG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 344 13 "Extremstellen" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 291 21 "Notwendige B edingung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "xfs:=solve(fs( x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xfsG$\"\"#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 292 23 "Hinreichende Bedingung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "fss(xfs);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+++++S! \"*" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 257 50 "Ergebnis > 0, dann ist \+ es ein lokales Minimum (TP)" }}{PARA 256 "" 0 "" {TEXT 256 50 "Ergebni s < 0, dann ist es ein lokales Maximum (HP)" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 293 12 "=> Hochpunkt" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(xfs);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 294 33 "Der Hochpunk t liegt bei HP(2 / 1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "p4 :=plot([[2,1]],x=-1..5,-5..2.1,style=point,colour=black,symbol=circle) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" } {TEXT 343 12 "Wendestellen" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 295 21 "Notwendige Bedingung:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "xfss:=solve(fss(x)=0,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xfssG6$$\"+++++:!\"*$\"+++++DF(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 296 23 "Hinreichende Bedingung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fsss(xfss[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+ _0\\/(*!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fsss(xfss[2] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+_0\\/(*!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 258 76 "Es gibt zwei Wendepunkte, da fsss(xfss[1]) und fsss(xfss[1]) u ngleich 0 ist." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f(xfss[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(f1`1'!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f(xfss[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(f 1`1'!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 297 69 "Die Wendepunkte liegen bei WP1( -9.7 / 0 .61 ) und WP2 ( 9.7 / 0.61 )." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "p5:=plot([[xfss[1],f(xfss[1])],[xfss[2],f(xfss[2])]],x=-1..5,-5 ..2.1,style=point,colour=black,symbol=circle):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 342 9 "Schaubild" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 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Die Dichte f \+ besitzt ein globales Maximum im Punkt x=mu, sowie zwei Wendepunkte. Mu ist der Mittelwert und sigma die Standardabweichung der Verteilung" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 312 366 "Die Normalverteilung ist eine wichtige Wahrscheinlic hkeitsverteilung. Man kann mit der Normalverteilung in vielerlei prakt ischen Anwendungen rechnen, dass zumindest n\344herungsweise die Verte ilungsgestalt einer Gauschen Glockenkurve vorliegt, beispielsweise dan n, wenn die Messung einer an sich festen Gr\366\337e beschreibt, wobei aber zuf\344llige Me\337fehler auftreten k\366nnen." }}{PARA 268 "" 0 "" {TEXT 313 0 "" }}{PARA 269 "" 0 "" {TEXT 269 15 "ZUR ERKL\304RUNG : " }{TEXT 314 493 "Eine Maschine produziert Stahlstifte mit einer Sol l-L\344nge von 35mm. W\344hrend der Herstellung k\366nnen Ungenauigkei ten auftreten, d.h. dass es auch mal l\344ngere oder k\374rzere Stahls tifte entstehen. Dadurch k\366nnen Abweichungen zur Soll-L\344nge, als o Me\337fehler, entstehen. Diese Wahrscheinlichkeitsverteilung kann mi t der Glockenkurve dargestellt werden. Mit der Normalverteilung kann m an z.B. berechnen, wie gro\337 die Wahrscheinlichkeit ist, dass die St ahlstifte zwischen 32 und 37 mm produziert werden." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 270 39 "Verdeutlichung anhand eines Beispieles:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 266 "" 0 " " {TEXT 315 112 "Eine H\344ufigkeitsverteilung f\374r die im Jahre 199 8 verkauften Gr\366\337en bei Herrenschuhen zeigt die folgende Tabelle :" }}{PARA 0 "" 0 "" {TEXT 316 158 "Gr\366\337en x[i]: 37 \+ 38 39 40 41 42 43 44 \+ 45 46 47 48 49" }}{PARA 0 "" 0 "" {TEXT 317 159 "--------------------------------------------- ---------------------------------------------------------------------- --------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 318 70 "Prozent p(x[i]) 0 0.2 1.2 3.8 1 1.8 " }{TEXT 265 5 "20.8 " }{TEXT 319 4 " " }{TEXT 264 7 " 24.5 " }{TEXT 320 67 " 17.4 10.5 7.9 1.2 \+ 0.6 0.1" }}{PARA 0 "" 0 "" {TEXT 321 0 "" }}{PARA 0 "" 0 "" {TEXT 271 164 "Es f\344llt auf, dass offensichtlich die meisten Herren die Schuhgr\366\337e 42 bis 43 haben. Das folgende Schaubild zeigt di e grafische Veranschaulichung dieser Verteilung." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "with(stats):with(statev alf):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(statplots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "f1:=statevalf[pdf,normald [43,1.71]]:plot(f1(x),x=36..49,-0.1..0.3,thickness=3,colour=blue);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6#7[o7 $$\"#O\"\"!$\"+0QPf`!#97$$\"++jLGO!\")$\"+::sT5!#87$$\"+k:*Hl$F1$\"+`a J;=F47$$\"+')*=2o$F1$\"+_Jx5LF47$$\"+/-j3PF1$\"+ir:+fF47$$\"+l(3kt$F1$ \"+2Y@@5!#77$$\"+YI;iPF1$\"+8%3(e;FI7$$\"+8,$))y$F1$\"+cV8wEFI7$$\"+K# 4k\"QF1$\"+X***zF%FI7$$\"+/***Q%QF1$\"+:O#Rl'FI7$$\"+Ru%oSF]o7$$\"+n#[]+%F1$\"+#y&oq_F]o7$$\"+= dMMSF1$\"+r3pzpF]o7$$\"+-X;fSF1$\"+rPQ`')F]o7$$\"+\\Y.)3%F1$\"+c+4#3\" !#57$$\"+N@\"35%F1$\"+%>&z$=\"Ffq7$$\"+?'*e8TF1$\"+e)[yG\"Ffq7$$\"+Z&3 w7%F1$\"+(Q>NS\"Ffq7$$\"+uuiTTF1$\"+oJL>:Ffq7$$\"+3o(\\:%F1$\"+7EGG;Ff q7$$\"+ThKoTF1$\"+6@WM%F1$\"+B?LS>Ffq7$$\"+e]w@UF1$\"+ :$o65#Ffq7$$\"+M$e$\\UF1$\"+>!**GB#Ffq7$$\"+j\"*ojUF1$\"+no)4G#Ffq7$$ \"+#**>!yUF1$\"+jA!QJ#Ffq7$$\"+2vD%G%F1$\"+'HIJK#Ffq7$$\"+@]\\!H%F1$\" +(\\%RHBFfq7$$\"+ODt'H%F1$\"+D*pDL#Ffq7$$\"+\\+(HI%F1$\"+=RkKBFfq7$$\" +Pnq4VF1$\"+7,CHBFfq7$$\"+BMW;VF1$\"+'HMAK#Ffq7$$\"+5,=BVF1$\"+(**e;J# Ffq7$$\"+&z;*HVF1$\"+/Kc(H#Ffq7$$\"+3h$QM%F1$\"+LQedAFfq7$$\"+?avdVF1$ \"+T0l.AFfq7$$\"+%3!*\\Q%F1$\"+3W#>1#Ffq7$$\"+A5M6WF1$\"+^^N()=Ffq7$$ \"+X#*fSWF1$\"+il%Qm\"Ffq7$$\"+`Tu`WF1$\"+>.Ld:Ffq7$$\"+g!*)oY%F1$\"+; ^/\\9Ffq7$$\"+bS#4[%F1$\"+YA+L8Ffq7$$\"+\\!f\\\\%F1$\"+7&>!=7Ffq7$$\"+ Ton2XF1$\"+cj!f6\"Ffq7$$\"+JYR?XF1$\"+BXr;5Ffq7$$\"+SC?[XF1$\"+<-RO\") F]o7$$\"+AuOuXF1$\"+-J7SkF]o7$$\"+ntr,YF1$\"+%>Q\">\\F]o7$$\"+PpXGYF1$ \"+_1p(o$F]o7$$\"+*y]kl%F1$\"+Y)zol#F]o7$$\"+'>7Mo%F1$\"+JT-*)=F]o7$$ \"+GT)4r%F1$\"+78#*)H\"F]o7$$\"+^xKQZF1$\"+*[2At)FI7$$\"++PXjZF1$\"+a5 PFfFI7$$\"+u3D#z%F1$\"+w1S-PFI7$$\"+5u+=[F1$\"+;)RFP#FI7$$\"+[#pa%[F1$ \"+H]-S9FI7$$\"+KPvr[F1$\"+0**f:()F47$$\"#\\F*$\"+#R9*[\\F4-%'COLOURG6 &%$RGBGF*F*$\"*++++\"F1-%+AXESLABELSG6$Q\"x6\"%!G-%*THICKNESSG6#\"\"$- %%VIEWG6$;F(F__l;$!\"\"Fi`l$Fb`lFi`l" 1 2 0 1 3 2 6 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:\nwith(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 144 "Anhand der Werte aus der Tabelle kann der Mittelwert mu und die Standardabweichung sigma, sowie die Funktion der Glockenkurve berechnet werden. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 63 "Der x-Wert des Hochpunktes ist der Mittelwert mu der Ver teilung" }}{PARA 0 "" 0 "" {TEXT 274 13 "Dabei ist mu:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "mu=1/100*Sum(x[i]*f(x[i]),i=1..13); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#muG,$-%$SumG6$*&&%\"xG6#%\"iG\" \"\"-%\"fG6#F*F./F-;F.\"#8#F.\"$+\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "mu:=1/ 100*(37*0 + 38*0.2 + 39*1.2 + 40*3.8 + 41*11.8 + 42*20.8 + 43*24.5 + 4 4* 17.4 + 45*10.5 + 46*7.9 + 47*1.2 + 48*0.6 + 49*0.1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#muG$\"+++!*3V!\")" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 275 111 "Der y-Wert des Hochpunktes \+ wird durch die Standardabweichung sigma bestimmt, ebenso wie die \"Bre ite der Glocke\"" }}{PARA 0 "" 0 "" {TEXT 276 16 "Dabei ist sigma:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "sigma^2=1/100*Sum((x[i]-mu) *f(x[i]),i=1..13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&sigmaG\"\" #\"\"\",$-%$SumG6$*&,&&%\"xG6#%\"iG\"\"\"$!+++!*3V!\")F3F3-%\"fG6#F/F3 /F2;F3\"#8#F3\"$+\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "sigma2:=1/100*((37-mu)^2*0 \+ + (38-mu)^2*0.2 + (39-mu)^2*1.2 + (40-mu)^2*3.8 + (41-mu)^2*11.8 + (42 -mu)^2*20.8 + (43-mu)^2*24.5 + (44-mu)^2*17.4 + (45-mu)^2*10.5 + (46-m u)^2*7.9 + (47-mu)^2*1.2 + (48-mu)^2*0.6 + (49-mu)^2*0.1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma2G$\"++!z!RH!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "sigma:=sqrt(sigma2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaG$\"+IUP9 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 330 51 "Sind noch zus\344tzlich die Bedingungen erf\374llt, dass " }}{PARA 0 "" 0 "" {TEXT -1 33 " " } {TEXT 331 68 " 68% der Messwerte zwischen +- 1*sigma vom Mittelwert mu aus liegen," }}{PARA 0 "" 0 "" {TEXT 332 94 " \+ 95% der Messwerte zwischen +- 2*sigma vom Mittelwert mu aus liegen, " }}{PARA 0 "" 0 "" {TEXT 333 96 " 99,7% der Messwerte zwischen +- 3*sigma vom Mittelwert mu aus liegen," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 334 77 "so kann man diese Daten durch e ine sogennante gau\337sche Glockenkurve ann\344hern." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 349 28 "\334berpr\374fung der Bedingungen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 350 0 "" }{TEXT 335 13 "1. Bedingung:" }{TEXT 336 68 " 68% der Messwer te liegen zwischen +- 1*sigma vom Mittelwert mu aus" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "mu-(1*sigma);mu+(1*sigma);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+xDYPT!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+BuL![%!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "20.8+24. 5+17.4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 337 13 "2. 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