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cleartomark
%%EndFont 
%%BeginFont: MSBM10
%!PS-AdobeFont-1.1: MSBM10 2.1
%%CreationDate: 1993 Sep 17 11:10:37
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 69 /E put
dup 82 /R put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 1 /periodcentered put
dup 20 /lessequal put
dup 21 /greaterequal put
dup 33 /arrowright put
dup 49 /infinity put
dup 50 /element put
dup 102 /braceleft put
dup 103 /braceright put
dup 106 /bar put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR7
%!PS-AdobeFont-1.1: CMR7 1.0
%%CreationDate: 1991 Aug 20 16:39:21
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 48 /zero put
dup 49 /one put
dup 50 /two put
readonly def
/FontBBox{-27 -250 1122 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMEX10
%!PS-AdobeFont-1.1: CMEX10 1.00
%%CreationDate: 1992 Jul 23 21:22:48
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMEX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMEX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 8 /braceleftbig put
dup 9 /bracerightbig put
dup 18 /parenleftbigg put
dup 19 /parenrightbigg put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891
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cleartomark
%%EndFont 
%%BeginFont: CMMI10
%!PS-AdobeFont-1.1: CMMI10 1.100
%%CreationDate: 1996 Jul 23 07:53:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 25 /pi put
dup 58 /period put
dup 59 /comma put
dup 60 /less put
dup 61 /slash put
dup 66 /B put
dup 67 /C put
dup 78 /N put
dup 80 /P put
dup 85 /U put
dup 86 /V put
dup 88 /X put
dup 89 /Y put
dup 90 /Z put
dup 97 /a put
dup 98 /b put
dup 101 /e put
dup 102 /f put
dup 105 /i put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 116 /t put
dup 118 /v put
dup 120 /x put
dup 121 /y put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR10
%!PS-AdobeFont-1.1: CMR10 1.00B
%%CreationDate: 1992 Feb 19 19:54:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 10 /Omega put
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 58 /colon put
dup 61 /equal put
dup 91 /bracketleft put
dup 93 /bracketright put
dup 97 /a put
dup 101 /e put
dup 102 /f put
dup 105 /i put
dup 109 /m put
dup 110 /n put
dup 112 /p put
dup 120 /x put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: SFBX1000
%!FontType1-1.0: SFBX1000 0.3
%%CreationDate: Wed Sep 12 2001
% Copyright (c) 2001 Vladimir Volovich <vvv@vsu.ru>.
% See the file COPYING (GNU General Public License) for license conditions.
% Converted from METAFONT EC/TC and LH fonts:
% ecbx1000, tcbx1000, labx1000, lbbx1000, lcbx1000, rxbx1000.
11 dict begin
/FontInfo 6 dict dup begin
/version (0.3) def
/FullName (Computer Modern Bold Extended) def
/FamilyName (Computer Modern) def
/ItalicAngle 0 def
/isFixedPitch false def
/Weight (Bold) def
end readonly def
/FontName /SFBX1000 def
/Encoding StandardEncoding def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] def
/FontBBox{-229 -316 1737 939}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: SFRM1440
%!FontType1-1.0: SFRM1440 0.3
%%CreationDate: Wed Sep 12 2001
% Copyright (c) 2001 Vladimir Volovich <vvv@vsu.ru>.
% See the file COPYING (GNU General Public License) for license conditions.
% Converted from METAFONT EC/TC and LH fonts:
% ecrm1440, tcrm1440, larm1440, lbrm1440, lcrm1440, rxrm1440.
11 dict begin
/FontInfo 6 dict dup begin
/version (0.3) def
/FullName (Computer Modern Roman) def
/FamilyName (Computer Modern) def
/ItalicAngle 0 def
/isFixedPitch false def
/Weight (Medium) def
end readonly def
/FontName /SFRM1440 def
/Encoding StandardEncoding def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] def
/FontBBox{-178 -319 1370 944}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: SFBX1440
%!FontType1-1.0: SFBX1440 0.3
%%CreationDate: Wed Sep 12 2001
% Copyright (c) 2001 Vladimir Volovich <vvv@vsu.ru>.
% See the file COPYING (GNU General Public License) for license conditions.
% Converted from METAFONT EC/TC and LH fonts:
% ecbx1440, tcbx1440, labx1440, lbbx1440, lcbx1440, rxbx1440.
11 dict begin
/FontInfo 6 dict dup begin
/version (0.3) def
/FullName (Computer Modern Bold Extended) def
/FamilyName (Computer Modern) def
/ItalicAngle 0 def
/isFixedPitch false def
/Weight (Bold) def
end readonly def
/FontName /SFBX1440 def
/Encoding StandardEncoding def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] def
/FontBBox{-218 -316 1652 915}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: SFRM1000
%!FontType1-1.0: SFRM1000 0.3
%%CreationDate: Wed Sep 12 2001
% Copyright (c) 2001 Vladimir Volovich <vvv@vsu.ru>.
% See the file COPYING (GNU General Public License) for license conditions.
% Converted from METAFONT EC/TC and LH fonts:
% ecrm1000, tcrm1000, larm1000, lbrm1000, lcrm1000, rxrm1000.
11 dict begin
/FontInfo 6 dict dup begin
/version (0.3) def
/FullName (Computer Modern Roman) def
/FamilyName (Computer Modern) def
/ItalicAngle 0 def
/isFixedPitch false def
/Weight (Medium) def
end readonly def
/FontName /SFRM1000 def
/Encoding StandardEncoding def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] def
/FontBBox{-189 -321 1456 937}readonly def
currentdict end
currentfile eexec
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TeXDict begin 1 0 bop 0 36 a Fl(Prof.)28 b(Dr.)f(V)-7
b(olk)n(er)27 b(Sc)n(hmidt)2303 b(WS)28 b(2008/09)0 135
y(Dipl.-Math.)g(o)r(ec.)f(Ralf)h(Thiedmann)2068 b(18.12.2008)0
235 y(Dipl.-Math.)28 b(o)r(ec.)f(Florian)g(V)-7 b(oss)259
531 y Fk(Elemen)l(tare)44 b(W)-11 b(ahrsc)l(heinlic)l(hk)l(eitsrec)l
(hn)l(ung)45 b(und)e(Statistik)1446 669 y Fj(\334bungsblatt)c(10)896
816 y Fl(\(Abgab)r(e:)28 b(Donnerstag,)e(8.01.2009,)e(v)n(or)i(den)i
(\334bungen\))0 1013 y Fi(Aufgab)s(e)k(1)145 b Fl(\(2)28
b(+)f(2)g(+)g(1)g(Punkte\))0 1136 y(Der)g(Zufallsv)n(ektor)f
Fh(\()p Fg(X)r(;)14 b(Y)19 b Fh(\))28 b Fl(b)r(esitze)f(die)h
(gemeinsame)f(Dic)n(h)n(te)490 1368 y Fg(f)9 b Fh(\()p
Fg(x;)14 b(y)s Fh(\))23 b(=)878 1312 y(1)p 853 1349 92
4 v 853 1425 a(2)p Fg(\031)968 1251 y Ff(\022)1030 1368
y Fh(1)18 b(+)g Fg(xy)s Fh(\()p Fg(x)1343 1334 y Fe(2)1399
1368 y Fd(\000)g Fg(y)1526 1334 y Fe(2)1563 1368 y Fh(\))c(exp)1750
1251 y Ff(\022)1811 1368 y Fd(\000)1886 1312 y Fh(1)p
1886 1349 42 4 v 1886 1425 a(2)1937 1368 y(\()p Fg(x)2016
1334 y Fe(2)2072 1368 y Fh(+)k Fg(y)2199 1334 y Fe(2)2236
1368 y Fh(\))2268 1251 y Ff(\023)q(\023)2409 1368 y Fd(\001)h
Fh(exp)2592 1251 y Ff(\022)2653 1368 y Fd(\000)2728 1312
y Fh(1)p 2728 1349 V 2728 1425 a(2)2779 1368 y(\()p Fg(x)2858
1334 y Fe(2)2914 1368 y Fh(+)f Fg(y)3041 1334 y Fe(2)3078
1368 y Fh(\))3110 1251 y Ff(\023)0 1606 y Fl(f\374r)28
b Fh(\()p Fg(x;)14 b(y)s Fh(\))23 b Fd(2)h Fc(R)486 1576
y Fe(2)523 1606 y Fl(.)60 1756 y(\(a\))42 b(Bestimmen)27
b(Sie)h(die)g(Randv)n(erteilungen)e(v)n(on)h Fg(X)34
b Fl(und)28 b Fg(Y)19 b Fl(.)55 1905 y(\(b\))43 b(Berec)n(hnen)26
b(Sie)i(den)g(K)n(orrelationsk)n(o)r(e\036zien)n(ten)23
b(v)n(on)k Fg(X)34 b Fl(und)28 b Fg(Y)19 b Fl(.)65 2054
y(\(c\))42 b(Sind)28 b Fg(X)34 b Fl(und)28 b Fg(Y)46
b Fl(unabh\344ngig?)0 2280 y Fi(Aufgab)s(e)32 b(2)145
b Fl(\(4)28 b(Punkte\))0 2403 y(Seien)35 b Fg(X)41 b
Fl(und)35 b Fg(Y)53 b Fl(unabh\344ngig)33 b(und)i(gleic)n(h)n(v)n
(erteilt)e(auf)i Fh([0)p Fg(;)14 b Fh(1])p Fl(.)34 b(Berec)n(hnen)f
(Sie)i(die)g(K)n(o)n(v)-5 b(arianzen)32 b Fg(C)6 b(ov)s
Fh(\()p Fg(X)r(;)14 b(Y)19 b Fh(\))0 2503 y Fl(und)28
b Fg(C)6 b(ov)s Fh(\()p Fg(U;)14 b(V)19 b Fh(\))p Fl(,)28
b(w)n(ob)r(ei)g Fg(U)j Fh(=)23 b(min)p Fd(f)p Fg(X)r(;)14
b(Y)k Fd(g)27 b Fl(und)h Fg(V)42 b Fh(=)23 b(max)o Fd(f)p
Fg(X)r(;)14 b(Y)k Fd(g)p Fl(.)0 2728 y Fi(Aufgab)s(e)32
b(3)145 b Fl(\(2)28 b(+)f(2)g(+)g(2)g(Punkte\))0 2852
y(Seien)h Fg(X)r(;)14 b(Y)5 b(;)14 b(Z)32 b Fl(quadratisc)n(h)26
b(in)n(tegrierbare)f(Zufallsv)-5 b(ariablen.)60 3001
y(\(a\))42 b(Zeigen)26 b(Sie)i(f\374r)g Fg(a;)14 b(b)22
b Fd(2)i Fc(R)p Fl(,)j(dass)208 3101 y Fg(C)6 b(ov)s
Fh(\()p Fg(aX)13 b Fh(+)7 b Fg(bY)e(;)14 b(Z)6 b Fh(\))22
b(=)g Fg(aC)6 b(ov)s Fh(\()p Fg(X)r(;)14 b(Z)6 b Fh(\))h(+)g
Fg(bC)f(ov)s Fh(\()p Fg(Y)f(;)14 b(Z)6 b Fh(\))22 b Fl(und)g
Fg(C)6 b(ov)s Fh(\()p Fg(Z)q(;)14 b(aX)f Fh(+)7 b Fg(bY)18
b Fh(\))23 b(=)g Fg(aC)6 b(ov)s Fh(\()p Fg(Z)q(;)14 b(X)7
b Fh(\))g(+)g Fg(bC)f(ov)s Fh(\()p Fg(Z)q(;)14 b(Y)k
Fh(\))p Fl(.)55 3250 y(\(b\))43 b(Zeigen)26 b(Sie)i(:)g(W)-7
b(enn)28 b Fg(X)34 b Fl(und)28 b Fg(Y)46 b Fl(iden)n(tisc)n(h)28
b(v)n(erteilt)f(sind,)g(dann)h(sind)g Fg(X)c Fd(\000)18
b Fg(Y)47 b Fl(und)28 b Fg(X)c Fh(+)19 b Fg(Y)46 b Fl(unk)n(orreliert.)
65 3399 y(\(c\))c(Seien)33 b Fg(X)39 b Fl(und)34 b Fg(Y)52
b Fl(unabh\344ngig)32 b(und)h Fg(B)t(in)p Fh(\(1)p Fg(;)14
b(p)p Fh(\))p Fl(-v)n(erteilt,)32 b Fh(0)g Fg(<)g(p)g(<)g
Fh(1)p Fl(.)h(Sind)h(dann)f Fg(X)28 b Fd(\000)22 b Fg(Y)52
b Fl(und)34 b Fg(X)28 b Fh(+)22 b Fg(Y)208 3499 y Fl(unabh\344ngig?)0
3725 y Fi(Aufgab)s(e)32 b(4)145 b Fl(\(5)28 b(Punkte\))0
3848 y(Laut)e(Bro)r(c)n(khaus)e(sind)i(4\045)g(der)g(m\344nnlic)n(hen)f
(Bundesb\374rger)g(Linksh\344nder.)g(Geb)r(en)i(Sie)f(mit)g(Hilfe)h
(der)f(Unglei-)0 3947 y(c)n(h)n(ung)31 b(v)n(on)g(T)-7
b(sc)n(heb)n(yc)n(hev)31 b(eine)h(Absc)n(h\344tzung)f(nac)n(h)g(un)n
(ten)i(f\374r)f(die)g(W)-7 b(ahrsc)n(heinlic)n(hk)n(eit)30
b(an,)i(dass)f(un)n(ter)h(800)0 4047 y(Studen)n(ten)c(zwisc)n(hen)f(24)
f(und)i(40)e(Linksh\344nder)h(sind.)g(V)-7 b(ergleic)n(hen)27
b(Sie)g(das)g(Ergebnis)g(mit)h(dem)f(w)n(ahren)f(W)-7
b(ert)0 4147 y Fh(0)p Fg(:)p Fh(8761)32 b Fl(un)n(ter)i(Binomialv)n
(erteilung)e(b)r(ezieh)n(ungsw)n(eise)h(dem)i(N\344herungsw)n(ert)d(un)
n(ter)i(Normalv)n(erteilung)f(\(zen-)0 4246 y(traler)g(Grenzw)n
(ertsatz)g(v)n(on)g(DeMoivre-Laplace\).)g(V)-7 b(erw)n(enden)34
b(Sie)g(f\374r)h(die)f(W)-7 b(erte)35 b(der)f(V)-7 b
(erteilungsfunktion)0 4346 y(der)27 b Fg(N)9 b Fh(\(0)p
Fg(;)14 b Fh(1\))p Fl(-V)-7 b(erteilung)27 b(die)g(Quan)n(til-T)-7
b(ab)r(elle)27 b(auf)g(der)h(Homepage.)0 4571 y Fi(Aufgab)s(e)k(5)145
b Fl(\(2)28 b(Punkte\))0 4695 y(Sei)g Fg(X)h Fh(:)23
b(\012)g Fd(!)g Fc(R)28 b Fl(eine)g(b)r(eliebige)f(Zufallsv)-5
b(ariable)26 b(mit)i Fc(E)14 b Fd(j)p Fg(X)7 b Fd(j)23
b Fg(<)g Fd(1)p Fl(.)k(Zeigen)g(Sie,)h(dass)f(f\374r)g(jedes)h
Fg(t)23 b Fd(\025)g Fh(0)1256 4877 y Fg(P)12 b Fh(\()p
Fg(X)29 b Fd(\025)23 b Fg(t)p Fh(\))g Fd(\024)34 b Fh(inf)1712
4931 y Fb(u)p Fa(\025)p Fe(0)1864 4810 y Ff(\010)1912
4877 y Fg(e)1951 4843 y Fa(\000)p Fb(tu)2085 4877 y Fc(E)14
b Fg(e)2204 4843 y Fb(uX)2307 4810 y Ff(\011)2383 4877
y Fg(:)0 5153 y Fi(Aufgab)s(e)32 b(6)145 b Fl(\(5)28
b(Punkte\))0 5277 y(Es)g(seien)f Fg(a)364 5289 y Fe(1)401
5277 y Fg(;)14 b(:)g(:)g(:)g(;)g(a)630 5289 y Fb(n)703
5277 y Fl(p)r(ositiv)n(e)27 b(Zahlen.)g(Bew)n(eisen)g(Sie)g(mittels)h
(der)g(Jensen-Ungleic)n(h)n(ung,)d(dass)1568 5459 y Fg(a)1612
5471 y Fb(H)1698 5459 y Fd(\024)d Fg(a)1829 5471 y Fb(G)1908
5459 y Fd(\024)h Fg(a)2040 5471 y Fb(A)0 5642 y Fl(w)n(ob)r(ei)33
b Fg(a)285 5654 y Fb(A)371 5642 y Fh(=)f(1)p Fg(=n)20
b Fd(\001)j Fh(\()p Fg(a)744 5654 y Fe(1)803 5642 y Fh(+)f
Fd(\001)14 b(\001)g(\001)22 b Fh(+)f Fg(a)1139 5654 y
Fb(n)1185 5642 y Fh(\))33 b Fl(das)f(arithmetisc)n(he)h(Mittel,)h
Fg(a)2248 5654 y Fb(G)2335 5642 y Fh(=)e(\()p Fg(a)2508
5654 y Fe(1)2568 5642 y Fd(\001)22 b(\001)14 b(\001)g(\001)22
b(\001)g Fg(a)2821 5654 y Fb(n)2866 5642 y Fh(\))2898
5612 y Fe(1)p Fb(=n)3044 5642 y Fl(das)32 b(geometrisc)n(he)0
5742 y(Mittel)c(und)g Fg(a)461 5754 y Fb(H)547 5742 y
Fh(=)23 b(\()681 5709 y Fe(1)p 677 5723 V 677 5770 a
Fb(n)728 5742 y Fh(\(1)p Fg(=a)888 5754 y Fe(1)943 5742
y Fh(+)18 b Fd(\001)c(\001)g(\001)k Fh(+)g(1)p Fg(=a)1352
5754 y Fb(n)1396 5742 y Fh(\)\))1460 5711 y Fa(\000)p
Fe(1)1578 5742 y Fl(das)27 b(harmonisc)n(he)f(Mittel)i(b)r(ezeic)n
(hnet.)p eop end
%%Trailer

userdict /end-hook known{end-hook}if
%%EOF