DERIVE for Windows version 5.02 DfW file saved on 30 Nov 2003 BINOMIAL_DENSITY(k, n, p):=COMB(n, k)p^k(1 - p)^(n - k) BINOMIAL_DISTRIBUTION(k, n, p):=(BINOMIAL_DENSITY(m_, n, p), m_, 0, MIN(k, n)) Bin_Den_Norm(k, n, p):=VECTOR([(k - mue(n, p))/sig(n, p), BINOMIAL_DENSITY(k, n, p)sig(n, p)], k, 0, n) CHI_SQUARE(x2, v):=INCOMPLETE_GAMMA(v/2, x2/2) DIGAMMA_PSI(z):=LN(z) - 1/(2z) + - 2INT(t_/((t_^2 + z^2)(^(2t_) - 1)), t_, 0, ) EULER_BETA(z, w):=(z)(w)/(z + w) F_DISTRIBUTION(f_, v1, v2):=INCOMPLETE_BETA(v2/(v2 + v1f_), v2/2, v1/2) HYPERGEOMETRIC_DENSITY(k, n, m, j):=COMB(m, k)COMB(j - m, n - k)/COMB(j, n) HYPERGEOMETRIC_DISTRIBUTION(k, n, m, j):=(HYPERGEOMETRIC_DENSITY(l_, n, m, j), l_, MAX(0, n - j + m), MIN(k, n, m)) INCOMPLETE_BETA(x, z, w):=(x^z/z + (1 - (1 - x)^w)/w + INT(t_^(z - 1)((1 - t_)^(w - 1) - 1) - (1 - t_)^(w - 1), t_, 0, x))/EULER_BETA(z, w) INCOMPLETE_GAMMA(z, w):=1/(z)(w^z/z + INT((^(-t_) - 1)t_^(z - 1), t_, 0, w)) INCOMPLETE_GAMMA_SERIES(z, w, m):=^(-w)w^z(w^(m - n_)/(z + m - n_ + 1), n_, 0, m) POCHHAMMER(a, x):=PERM(x + a + -1, x) POISSON_DENSITY(k, t):=^(-t)t^k/k! POISSON_DISTRIBUTION(k, t):=(POISSON_DENSITY(m_, t), m_, 0, k) POLYGAMMA(n, z):=IF(n = -1, LN((z)), IF(n = 0, DIGAMMA(z), (-1)^(n + 1)n!(n + 1, z))) SAW(n, p):=(np(1 - p))^(1/2) STUDENT(t, v):=1 - INCOMPLETE_BETA(v/(v + t^2), v/2, 1/2) mue(n, p):=np sig(n, p):=(np(1 - p))^(1/2) f_:= hCross:=APPROX(- 27/4) v1:= v2:= vCross:=APPROX(44500000000000001/100000000000000000) x2:= CTextObj 4{\rtf1\ansi\ansicpg1252\deff0\deflang1031{\fonttbl{\f0\fnil Arial Narrow;}{\f1\fnil\fcharset2 DfW5 Printer;}} \viewkind4\uc1\pard\b\f0\fs26 Standardisierung von Binomialverteilungen \par \f1 \par } @T{\rtf1\ansi\ansicpg1252\deff0\deflang1031{\fonttbl{\f0\fnil\fcharset0 Arial Narrow;}{\f1\fnil\fcharset2 DfW5 Printer;}} \viewkind4\uc1\pard\f0\fs22 mue und sigma werden zun\'e4chst festgelegt:\f1\fs24 \par } CExpnObj8`lBenutzer mue(n,p):=n*p8xBenutzer mue(300,0.6)Simp(#2)1808 Benutzersig(n,p):=(n*p*(1-p))^(0.5)8Benutzer sig(300,0.6)Simp(#5) 6*SQRT(2){\rtf1\ansi\ansicpg1252\deff0\deflang1031{\fonttbl{\f0\fswiss\fprq2\fcharset0 Arial Narrow;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs22 Standardisierung von b(k, 20, 0.6): \par } 8(BenutzerBINOMIAL_DENSITY(k,20,0.6)4hd Approx(#7)'2.675*10^10*#e^(0.40546*k)/(k!*(20-k)!)CPlotObjpfH C2DPlotView CExplicitPlotPU4@KBUB7yAB @U @ UC@K@ nDAAxy@?$@333333???11??BMv06(@0ThQ{\rtf1\ansi\ansicpg1252\deff0\deflang1031{\fonttbl{\f0\fswiss\fprq2\fcharset0 Arial Narrow;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs22 Die BINOMIAL_DENSITY - Werte werden mit Hilfe von mue und sig standardisiert. Um die Ergebnisse plotten zu k\'f6nnen, wird eine parametrisierte Darstellung gew\'e4hlt \par } 8t8Benutzer [Bin_Den_Norm(k,n,p):=VECTOR([(k-mue(n,p))/sig(n,p),BINOMIAL_DENSITY(k,n,p)*sig(n,p)],k,0,n)8Benutzer Bin_Den_Norm(k,20,0.6)p Approx(#12)Mb? [[-5.4772,2.4089*10^(-8)],[-5.0207,7.2267*10^(-7)],[-4.5643,1.0298*10^(-5)],[-4.1079,9.2682*10^(-5)],[-3.6514,0.00059085],[-3.1950,0.0028360],[-2.7386,0.010635],[-2.2821,0.031906],[-1.8257,0.077770],[-1.3693,0.15554],[-0.91287,0.25664],[-0.45643,0.34996],[0,0.39371],[0.45643,0.36342],[0.91287,0.27257],[1.3693,0.16354],[1.8257,0.076660],[2.2821,0.027056],[2.7386,0.0067642],[3.1950,0.0010680],[3.6514,8.0102*10^(-5)]]dPU4@KBUB7yAB @U @ UC@K@ nDAACPointListPlot >@PPB>@@A (@>@&@P@xB>@hA @>@@P@xB>@`A @>@@P@xB>@ A @>@@P@xB>@)B (@>@@PPB>@XLB@>@P@xB>@ŽEB (@>@@P@xB>@+`B@>@P@xB>@T}sB@>@P@xB>@T}B@>@PPB>@>!RB(@>@P@xB>@ +B @xB>@PnB(@>@@xB>@jRB@>@@xB>@CB@>@PB>@B@>@@xB>@}lsB(@>@@@xB>@k[B@>@@xB>@k;B(@>@@@xB>@QB@>@@PB>@ rA1U4@KBUBj@U@@nBxy@?@? > ףp=?11??BM.6(.{\rtf1\ansi\ansicpg1252\deff0\deflang1031{\fonttbl{\f0\fswiss\fprq2\fcharset0 Arial Narrow;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs22 In einzelnen Schritten: \par } 8Benutzer 3VECTOR([k,BINOMIAL_DENSITY(k,100,p)],p,0.1,0.9,0.1)f CParametricPlotU DUY@KBUB&KU_@K@@ m.-"oRY@k(G@"U DUY@KBUBJU@r@K@ / jo0WY@k(G@"U PUY@KBUB&KU]@K@ H@U @ s{OY@k(G@"U PUY@KBUBJUb@K@ @X@U @ m.-"oRY@k(G@"U $UY@KBUB @ܕN0ZY@k(G@"U PUY@KBUBJH@U @ Uh@K@ m.-"oRY@k(G@"U PUY@KBUB&K0@U @ Ub@K@ s{OY@k(G@"U DUY@KBUBJ@X@U@ @ / jo0WY@k(G@"U DUY@KBUB&K8@U @@ m.-"oRY@k(G@xy $@?I@333333? M@?11??BMv06(@0{\rtf1\ansi\ansicpg1252\deff0\deflang1031{\fonttbl{\f0\fswiss\fprq2\fcharset0 Arial Narrow;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs22 Verschiebung um mue(n, p) f\'fchrt zu Erwartungswert = 0 bei ALLEN Verteilungen. \par } 8xBenutzer >VECTOR([k-mue(100,p),BINOMIAL_DENSITY(k,100,p)],p,0.1,0.9,0.1)*f"U DUY@KBUB&KU_@K@@ m.-"oRY@k(G@"U DUY@KBUBJU@r@K@ / jo0WY@k(G@"U PUY@KBUB&KU]@K@ H@U @ s{OY@k(G@"U PUY@KBUBJUb@K@ @X@U @ m.-"oRY@k(G@"U $UY@KBUB @ܕN0ZY@k(G@"U PUY@KBUBJH@U @ Uh@K@ m.-"oRY@k(G@"U PUY@KBUB&K0@U @ Ub@K@ s{OY@k(G@"U DUY@KBUBJ@X@U@ @ / jo0WY@k(G@"U DUY@KBUB&K8@U @@ m.-"oRY@k(G@" $@UK DUY@KBUB&KU_@K@@ m.-"oR? 5P?" 4@UK DUY@KBUBJU@r@K@ / jo0W? 5P?" >@UK PUY@KBUB&KU]@K@ H@U @ s{O? 5P?" D@UK PUY@KBUBJUb@K@ @X@U @ m.-"oR? 5P?" I@UK $UY@KBUB @ܕN0Z? 5P?" N@UK PUY@KBUBJH@U @ Uh@K@ m.-"oR? 5P?" Q@UK PUY@KBUB&K0@U @ Ub@K@ s{O? 5P?" T@UK DUY@KBUBJ@X@U@ @ / jo0W? 5P?" V@UK DUY@KBUB&K8@U @@ m.-"oR? 5P?" $@UK DUY@KBUB&KU_@K@@ m.-"oRY@k(G@" 4@UK DUY@KBUBJU@r@K@ / jo0WY@k(G@" >@UK PUY@KBUB&KU]@K@ H@U @ s{OY@k(G@" D@UK PUY@KBUBJUb@K@ @X@U @ m.-"oRY@k(G@" I@UK $UY@KBUB @ܕN0ZY@k(G@" N@UK PUY@KBUBJH@U @ Uh@K@ m.-"oRY@k(G@" Q@UK PUY@KBUB&K0@U @ Ub@K@ s{OY@k(G@" T@UK DUY@KBUBJ@X@U@ @ / jo0WY@k(G@" V@UK DUY@KBUB&K8@U @@ m.-"oRY@k(G@xy4@?D@333333?I$I$M@?11??BM-6(- !{\rtf1\ansi\ansicpg1252\deff0\deflang1031{\fonttbl{\f0\fswiss\fprq2\fcharset0 Arial Narrow;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs22 Einschr\'e4nkung auf die neuen Graphen: \par } -f  " $@UK DUY@KBUB&KU_@K@@ m.-"oRY@k(G@" 4@UK DUY@KBUBJU@r@K@ / jo0WY@k(G@" >@UK PUY@KBUB&KU]@K@ H@U @ s{OY@k(G@" D@UK PUY@KBUBJUb@K@ @X@U @ m.-"oRY@k(G@" I@UK $UY@KBUB @ܕN0ZY@k(G@" N@UK PUY@KBUBJH@U @ Uh@K@ m.-"oRY@k(G@" Q@UK PUY@KBUB&K0@U @ Ub@K@ s{OY@k(G@" T@UK DUY@KBUBJ@X@U@ @ / jo0WY@k(G@" V@UK DUY@KBUB&K8@U @@ m.-"oRY@k(G@xy@??[kZ6i?11??BMv06(@0 % {\rtf1\ansi\ansicpg1252\deff0\deflang1031{\fonttbl{\f0\fswiss\fprq2\fcharset0 Arial Narrow;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs22 Anpassen der Breite: \par } 81 U BenutzerKVECTOR([(k-mue(100,p))/sig(100,p),BINOMIAL_DENSITY(k,100,p)],p,0.1,0.9,0.1)a f9  "@$@UKDUY@KBUB&KU_@K@@ m.-"oRY@k(G@"@4@UKDUY@KBUBJU@r@K@ / jo0WY@k(G@"6@5@$@5@U5@KPUY@KBUB&KU]@K@ H@U @ s{OY@k(G@"6@@$@(@U@KPUY@KBUBJUb@K@ @X@U @ m.-"oRY@k(G@"@I@UK$UY@KBUB @ܕN0ZY@k(G@",@@(@U@KPUY@KBUBJH@U @ Uh@K@ m.-"oRY@k(G@"6@5@$@5@U5@KPUY@KBUB&K0@U @ Ub@K@ s{OY@k(G@"@T@UKDUY@KBUBJ@X@U@ @ / jo0WY@k(G@"@V@UKDUY@KBUB&K8@U @@ m.-"oRY@k(G@xy ???UUUUUU6i?11??BMv06(@0E Y {\rtf1\ansi\ansicpg1252\deff0\deflang1031{\fonttbl{\f0\fswiss\fprq2\fcharset0 Arial Narrow;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs22 Anpassen der H\'f6he: \par } 8e  BenutzerVVECTOR([(k-mue(100,p))/sig(100,p),sig(100,p)*BINOMIAL_DENSITY(k,100,p)],p,0.1,0.9,0.1) fl  "@$@UKDUY@KBUB&KU@ o@K@ m.-"oRY@k(G@"@4@UKDUY@KBUBJU@r@K@ / jo0WY@k(G@"6@5@$@5@U5@KFUY@KBUB&KUP@@ 5@i4]hY@k(G@"6@@$@(@U@KFUY@KBUBJUP@@ @Rji.7gY@k(G@"@I@UK$UY@KBUB @=:]ZY@k(G@",@@(@U@KEUY@KBUBJU@@ @/ jocY@k(G@"6@5@$@5@U5@KEUY@KBUB&KU@@ 5@Rji.aY@k(G@"@T@UKDUY@KBUBJX@U@ @ / jo0WY@k(G@"@V@UKDUY@KBUB&KH@U@ @ m.-"oRY@k(G@xy ???UUUUUUffffff?11??BM-6(-x  {\rtf1\ansi\ansicpg1252\deff0\deflang1031{\fonttbl{\f0\fswiss\fprq2\fcharset0 Arial Narrow;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs22 Ende. \par }