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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "#\n# Kontroll-Karte
n:\n# a) Berechnung von gamma_n; E(S)=gamma_n*sigma\n# b) Berechnung v
on an = E(max(Z1,...,Zn))\n# c) Ze = Zentralwert von Z1,...,Zn; Berech
nung von Var(Zn)\n# d) Verteilungsfunktion der Spannweite R\n#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 31 "Phi:=z->1/2+1/2*erf(z/sqrt(2));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Phic:=z->(1/2)*erfc(z/sqrt(2));" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "phi:=z->exp(-z^2/2)/sqrt(2*
Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "#\n# Entwicklung zu
 gamma_n\n#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "gamma_n:=sqr
t(2/(n-1))*GAMMA(n/2)/GAMMA((n-1)/2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "limit(1-gamma_n^2,n=infinity);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 34 "limit(n*(1-gamma_n^2),n=infinity);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "#\n# Normal Scores: an = E(max(Z1,.
..,Zn)\n#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n:=100;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "an:=Int(x*n*Phi(x)^(n-1)*phi
(x),x=-infinity..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "#\n# Vertei
lung des Zentralwerts X = Ze(Z1,...,Zn) = Z(k+1) falls n=2k+1;\n#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "# F = Verteilungsfkt von X" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "k:=50;" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 9 "n:=2*k+1;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "F:=x->sum(binomial(n,j)*Phi(x)^j*Phic(x)^(n-j),j=k+1.
.n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=x->diff(F(x),x);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 45 "varianz:=Int(x^2*f(x),x=-infinity..infini
ty);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "n*%;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "evalf(Pi/2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 104 "#\n# Verteilungsfunktion der Spannweite\n# R = Z(n)-
Z(1), wobei Z(1)=min(Z1,...,Zn), Z(n)=max(Z1,...,Zn)\n#" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 70 "G := r -> n*int((Phi(x+r)-Phi(x))^(n-1)*phi(x)
,x=-infinity..infinity);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "n := 5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "
evalf(G(3.9));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "r0 := fso
lve(G(r)=0.95,r=3.8..3.9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "evalf(G(r0));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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