Ferdig for dagen

diff --git a/rapport.aux b/rapport.aux
index 5f0960fb31a8bfb04583391c10e1c6394a51a3e5..495181397ab14a48ab66d3ff09afe9936dd245c1 100644 (file)
--- a/rapport.aux
+++ b/rapport.aux
@@ -27,31 +27,31 @@
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-\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces \sf  Avg. number of jumps before ending up in state $100$, over $5000$ runs}}{7}}
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-\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces \sf  Ratio of runs that ended up in state $100$, over $5000$ runs}}{7}}
-\newlabel{tab:ratio_to_100}{{2}{7}}
+\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces \sf  Avg. number of jumps before ending up in state $100$, over $5000$ runs}}{6}}
+\newlabel{tab:jumps_to_100}{{1}{6}}
+\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces \sf  Ratio of runs that ended up in state $100$, over $5000$ runs}}{6}}
+\newlabel{tab:ratio_to_100}{{2}{6}}
+\@writefile{toc}{\contentsline {subsubsection}{Confidence intervals}{7}}
 \newlabel{eq:samplevariance}{{24}{7}}
 \newlabel{eq:z}{{25}{7}}
 \newlabel{eq:conf1}{{26}{7}}
 \newlabel{eq:conf1num}{{27}{7}}
+\newlabel{eq:z2}{{28}{7}}
+\newlabel{eq:conf2}{{29}{7}}
 \@writefile{lot}{\contentsline {table}{\numberline {3}{\ignorespaces \sf  Ratio of runs that ended up in state $100$ after $1000$ steps or more}}{8}}
 \newlabel{tab:ratio_to_100_after_1000}{{3}{8}}
-\newlabel{eq:z2}{{28}{8}}
-\newlabel{eq:conf2}{{29}{8}}
 \newlabel{eq:conf2num}{{30}{8}}
 \@writefile{toc}{\contentsline {subsection}{e)}{8}}
 \@writefile{toc}{\contentsline {section}{Problem 2}{8}}
 \@writefile{toc}{\contentsline {subsection}{a)}{8}}
-\@writefile{toc}{\contentsline {subsection}{b)}{9}}
+\@writefile{toc}{\contentsline {subsection}{b)}{8}}
 \@writefile{toc}{\contentsline {subsection}{c)}{9}}
 \newlabel{mat:small}{{31}{9}}
 \newlabel{eq:limprob01}{{32}{9}}
-\newlabel{eq:ui}{{33}{10}}
-\newlabel{eq:u2}{{34}{10}}
+\newlabel{eq:ui}{{33}{9}}
+\newlabel{eq:u2}{{34}{9}}
 \newlabel{eq:u3}{{35}{10}}
 \@writefile{toc}{\contentsline {subsection}{d)}{10}}
-\@writefile{lol}{\contentsline {lstlisting}{../Downloads/mostimportant.py}{11}}
-\newlabel{eq:lmatrix}{{37}{12}}
-\newlabel{eq:msim}{{38}{12}}
+\@writefile{lol}{\contentsline {lstlisting}{mostimportant.py}{10}}
+\newlabel{eq:lmatrix}{{37}{11}}
+\newlabel{eq:msim}{{38}{11}}

diff --git a/rapport.dvi b/rapport.dvi
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diff --git a/rapport.log b/rapport.log
index b290a4a79565bd03417fa31b33e78371fa8dd8d6..476cc680b18d7c8b0d477dded8efa2818fadca3a 100644 (file)
--- a/rapport.log
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diff --git a/rapport.pdf b/rapport.pdf
index baf67b02ad60d1e0c3efeb2bea757f9ba8c4dc87..68bd13c9d4fcfebe5f79475e145822ca72306864 100644 (file)
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diff --git a/rapport.ps b/rapport.ps
index 4e605c5a2760ef9b1b643c4013d7d3da41d1f89b..ec27d4de61d12d73c81442383ac2e8a00c736512 100644 (file)
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-%%Title: CMSS12
-%Version: 003.002
-%%CreationDate: Mon Jul 13 16:17:00 2009
-%%Creator: David M. Jones
-%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
-%Copyright: (<http://www.ams.org>), with Reserved Font Name CMSS12.
-% This Font Software is licensed under the SIL Open Font License, Version 1.1.
-% This license is in the accompanying file OFL.txt, and is also
-% available with a FAQ at: http://scripts.sil.org/OFL.
-%%EndComments
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-/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMSS12.) readonly def
-/FullName (CMSS12) readonly def
-/FamilyName (Computer Modern) readonly def
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-%%BeginFont: CMBX12
-%!PS-AdobeFont-1.0: CMBX12 003.002
-%%Title: CMBX12
-%Version: 003.002
-%%CreationDate: Mon Jul 13 16:17:00 2009
-%%Creator: David M. Jones
-%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
-%Copyright: (<http://www.ams.org>), with Reserved Font Name CMBX12.
-% This Font Software is licensed under the SIL Open Font License, Version 1.1.
-% This license is in the accompanying file OFL.txt, and is also
-% available with a FAQ at: http://scripts.sil.org/OFL.
-%%EndComments
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-/version (003.002) readonly def
-/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMBX12.) readonly def
-/FullName (CMBX12) readonly def
-/FamilyName (Computer Modern) readonly def
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-%%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.
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-%%Title: CMTI10
-%Version: 003.002
-%%CreationDate: Mon Jul 13 16:17:00 2009
-%%Creator: David M. Jones
-%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
-%Copyright: (<http://www.ams.org>), with Reserved Font Name CMTI10.
-% This Font Software is licensed under the SIL Open Font License, Version 1.1.
-% This license is in the accompanying file OFL.txt, and is also
-% available with a FAQ at: http://scripts.sil.org/OFL.
-%%EndComments
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-/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMTI10.) readonly def
-/FullName (CMTI10) readonly def
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-%%Title: CMMI10
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-%%CreationDate: Mon Jul 13 16:17:00 2009
-%%Creator: David M. Jones
-%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
-%Copyright: (<http://www.ams.org>), with Reserved Font Name CMMI10.
-% This Font Software is licensed under the SIL Open Font License, Version 1.1.
-% This license is in the accompanying file OFL.txt, and is also
-% available with a FAQ at: http://scripts.sil.org/OFL.
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-/UniqueID get 5087385 eq exch/FontType get 1 eq and}{pop false}ifelse
-{save true}{false}ifelse}{false}ifelse
-11 dict begin
-/FontType 1 def
-/FontMatrix [0.001 0 0 0.001 0 0 ]readonly def
-/FontName /CMMI10 def
-/FontBBox {-32 -250 1048 750 }readonly def
-/UniqueID 5087385 def
-/PaintType 0 def
-/FontInfo 10 dict dup begin
-/version (003.002) readonly def
-/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMMI10.) readonly def
-/FullName (CMMI10) readonly def
-/FamilyName (Computer Modern) readonly def
-/Weight (Medium) readonly def
-/ItalicAngle -14.04 def
-/isFixedPitch false def
-/UnderlinePosition -100 def
-/UnderlineThickness 50 def
-/ascent 750 def
-end readonly def
-/Encoding 256 array
-0 1 255 {1 index exch /.notdef put} for
-dup 22 /mu put
-dup 25 /pi put
-dup 58 /period put
-dup 59 /comma put
-dup 60 /less put
-dup 61 /slash put
-dup 62 /greater put
-dup 67 /C put
-dup 69 /E put
-dup 78 /N put
-dup 80 /P put
-dup 83 /S put
-dup 84 /T put
-dup 88 /X put
-dup 90 /Z put
-dup 97 /a put
-dup 100 /d put
-dup 105 /i put
-dup 106 /j put
-dup 107 /k put
-dup 109 /m put
-dup 110 /n put
-dup 112 /p put
-dup 113 /q put
-dup 114 /r put
-dup 117 /u put
-dup 118 /v put
-dup 122 /z put
-readonly def
-currentdict end
-currentfile eexec
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 {restore}if
 %%EndFont 
-%%BeginFont: CMSY10
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-%%Title: CMSY10
+%%BeginFont: CMMI12
+%!PS-AdobeFont-1.0: CMMI12 003.002
+%%Title: CMMI12
 %Version: 003.002
 %%CreationDate: Mon Jul 13 16:17:00 2009
 %%Creator: David M. Jones
 %Copyright: Copyright (c) 1997, 2009 American Mathematical Society
-%Copyright: (<http://www.ams.org>), with Reserved Font Name CMSY10.
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMMI12.
 % This Font Software is licensed under the SIL Open Font License, Version 1.1.
 % This license is in the accompanying file OFL.txt, and is also
 % available with a FAQ at: http://scripts.sil.org/OFL.
 %%EndComments
-FontDirectory/CMSY10 known{/CMSY10 findfont dup/UniqueID known{dup
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@@ -2663,32 +1097,32 @@ A96EFFACC0EFDE0292BBAAE6E5800814AC89848B7D709EFC133E4F1506106F
 cleartomark
 {restore}if
 %%EndFont 
-%%BeginFont: CMR10
-%!PS-AdobeFont-1.0: CMR10 003.002
-%%Title: CMR10
+%%BeginFont: CMSS12
+%!PS-AdobeFont-1.0: CMSS12 003.002
+%%Title: CMSS12
 %Version: 003.002
 %%CreationDate: Mon Jul 13 16:17:00 2009
 %%Creator: David M. Jones
 %Copyright: Copyright (c) 1997, 2009 American Mathematical Society
-%Copyright: (<http://www.ams.org>), with Reserved Font Name CMR10.
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMSS12.
 % This Font Software is licensed under the SIL Open Font License, Version 1.1.
 % This license is in the accompanying file OFL.txt, and is also
 % available with a FAQ at: http://scripts.sil.org/OFL.
 %%EndComments
-FontDirectory/CMR10 known{/CMR10 findfont dup/UniqueID known{dup
-/UniqueID get 5000793 eq exch/FontType get 1 eq and}{pop false}ifelse
+FontDirectory/CMSS12 known{/CMSS12 findfont dup/UniqueID known{dup
+/UniqueID get 5000804 eq exch/FontType get 1 eq and}{pop false}ifelse
 {save true}{false}ifelse}{false}ifelse
 11 dict begin
 /FontType 1 def
 /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def
-/FontName /CMR10 def
-/FontBBox {-40 -250 1009 750 }readonly def
-/UniqueID 5000793 def
+/FontName /CMSS12 def
+/FontBBox {-62 -251 978 758 }readonly def
+/UniqueID 5000804 def
 /PaintType 0 def
 /FontInfo 9 dict dup begin
 /version (003.002) readonly def
-/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMR10.) readonly def
-/FullName (CMR10) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMSS12.) readonly def
+/FullName (CMSS12) readonly def
 /FamilyName (Computer Modern) readonly def
 /Weight (Medium) readonly def
 /ItalicAngle 0 def
@@ -2698,56 +1132,12 @@ FontDirectory/CMR10 known{/CMR10 findfont dup/UniqueID known{dup
 end readonly def
 /Encoding 256 array
 0 1 255 {1 index exch /.notdef put} for
-dup 11 /ff put
-dup 12 /fi put
-dup 22 /macron put
-dup 33 /exclam put
-dup 34 /quotedblright put
-dup 37 /percent put
-dup 39 /quoteright put
-dup 40 /parenleft put
-dup 41 /parenright put
-dup 43 /plus put
 dup 44 /comma put
-dup 45 /hyphen put
 dup 46 /period put
-dup 47 /slash put
-dup 48 /zero put
-dup 49 /one put
-dup 50 /two put
-dup 51 /three put
-dup 52 /four put
-dup 53 /five put
-dup 54 /six put
-dup 55 /seven put
-dup 56 /eight put
-dup 57 /nine put
-dup 58 /colon put
-dup 61 /equal put
 dup 65 /A put
-dup 66 /B put
-dup 67 /C put
-dup 69 /E put
-dup 70 /F put
-dup 71 /G put
-dup 72 /H put
-dup 73 /I put
-dup 76 /L put
-dup 77 /M put
-dup 78 /N put
-dup 79 /O put
-dup 80 /P put
 dup 82 /R put
-dup 83 /S put
-dup 84 /T put
-dup 85 /U put
-dup 87 /W put
-dup 91 /bracketleft put
-dup 93 /bracketright put
-dup 94 /circumflex put
 dup 97 /a put
 dup 98 /b put
-dup 99 /c put
 dup 100 /d put
 dup 101 /e put
 dup 102 /f put
@@ -2755,682 +1145,281 @@ dup 103 /g put
 dup 104 /h put
 dup 105 /i put
 dup 106 /j put
-dup 107 /k put
-dup 108 /l put
 dup 109 /m put
 dup 110 /n put
 dup 111 /o put
 dup 112 /p put
-dup 113 /q put
 dup 114 /r put
 dup 115 /s put
-dup 116 /t put
-dup 117 /u put
-dup 118 /v put
-dup 119 /w put
-dup 120 /x put
-dup 121 /y put
-dup 122 /z put
-readonly def
-currentdict end
-currentfile eexec
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 cleartomark
 {restore}if
 %%EndFont 
-%%BeginFont: CMBX10
-%!PS-AdobeFont-1.0: CMBX10 003.002
-%%Title: CMBX10
+%%BeginFont: SFRM0800
+%!FontType1-1.0: SFRM0800 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:
+% ecrm0800, tcrm0800, larm0800, lbrm0800, lcrm0800, rxrm0800.
+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 /SFRM0800 def
+/Encoding StandardEncoding def
+/PaintType 0 def
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] def
+/FontBBox{-203 -320 1554 938}readonly def
+currentdict end
+currentfile eexec
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+cleartomark
+%%EndFont 
+%%BeginFont: CMTI8
+%!PS-AdobeFont-1.0: CMTI8 003.002
+%%Title: CMTI8
 %Version: 003.002
 %%CreationDate: Mon Jul 13 16:17:00 2009
 %%Creator: David M. Jones
 %Copyright: Copyright (c) 1997, 2009 American Mathematical Society
-%Copyright: (<http://www.ams.org>), with Reserved Font Name CMBX10.
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMTI8.
 % This Font Software is licensed under the SIL Open Font License, Version 1.1.
 % This license is in the accompanying file OFL.txt, and is also
 % available with a FAQ at: http://scripts.sil.org/OFL.
 %%EndComments
-FontDirectory/CMBX10 known{/CMBX10 findfont dup/UniqueID known{dup
-/UniqueID get 5000768 eq exch/FontType get 1 eq and}{pop false}ifelse
+FontDirectory/CMTI8 known{/CMTI8 findfont dup/UniqueID known{dup
+/UniqueID get 5000826 eq exch/FontType get 1 eq and}{pop false}ifelse
 {save true}{false}ifelse}{false}ifelse
 11 dict begin
 /FontType 1 def
 /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def
-/FontName /CMBX10 def
-/FontBBox {-56 -250 1164 750 }readonly def
-/UniqueID 5000768 def
+/FontName /CMTI8 def
+/FontBBox {-35 -250 1190 750 }readonly def
+/UniqueID 5000826 def
 /PaintType 0 def
 /FontInfo 9 dict dup begin
 /version (003.002) readonly def
-/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMBX10.) readonly def
-/FullName (CMBX10) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMTI8.) readonly def
+/FullName (CMTI8) readonly def
 /FamilyName (Computer Modern) readonly def
-/Weight (Bold) readonly def
-/ItalicAngle 0 def
+/Weight (Medium) readonly def
+/ItalicAngle -14.04 def
 /isFixedPitch false def
 /UnderlinePosition -100 def
 /UnderlineThickness 50 def
 end readonly def
 /Encoding 256 array
 0 1 255 {1 index exch /.notdef put} for
-dup 65 /A put
-dup 76 /L put
-dup 80 /P put
+dup 35 /numbersign put
 dup 97 /a put
 dup 98 /b put
 dup 99 /c put
-dup 100 /d put
 dup 101 /e put
-dup 102 /f put
-dup 104 /h put
-dup 105 /i put
-dup 107 /k put
-dup 108 /l put
-dup 109 /m put
-dup 110 /n put
+dup 103 /g put
 dup 111 /o put
 dup 112 /p put
-dup 114 /r put
-dup 115 /s put
-dup 116 /t put
-dup 117 /u put
-dup 119 /w put
+dup 118 /v put
 readonly def
 currentdict end
 currentfile eexec
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+%%EndFont 
+%%BeginFont: CMBX8
+%!PS-AdobeFont-1.0: CMBX8 003.002
+%%Title: CMBX8
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMBX8.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% available with a FAQ at: http://scripts.sil.org/OFL.
+%%EndComments
+FontDirectory/CMBX8 known{/CMBX8 findfont dup/UniqueID known{dup
+/UniqueID get 5000766 eq exch/FontType get 1 eq and}{pop false}ifelse
+{save true}{false}ifelse}{false}ifelse
+11 dict begin
+/FontType 1 def
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+/FontName /CMBX8 def
+/FontBBox {-59 -250 1235 750 }readonly def
+/UniqueID 5000766 def
+/PaintType 0 def
+/FontInfo 9 dict dup begin
+/version (003.002) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMBX8.) readonly def
+/FullName (CMBX8) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Bold) readonly def
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+/UnderlineThickness 50 def
+end readonly def
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+%%EndFont 
+%%BeginFont: CMTI10
+%!PS-AdobeFont-1.0: CMTI10 003.002
+%%Title: CMTI10
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMTI10.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% available with a FAQ at: http://scripts.sil.org/OFL.
+%%EndComments
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+{save true}{false}ifelse}{false}ifelse
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+/version (003.002) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMTI10.) readonly def
+/FullName (CMTI10) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
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+end readonly def
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-cleartomark
-{restore}if
-%%EndFont 
-%%BeginFont: CMSY8
-%!PS-AdobeFont-1.0: CMSY8 003.002
-%%Title: CMSY8
-%Version: 003.002
-%%CreationDate: Mon Jul 13 16:17:00 2009
-%%Creator: David M. Jones
-%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
-%Copyright: (<http://www.ams.org>), with Reserved Font Name CMSY8.
-% This Font Software is licensed under the SIL Open Font License, Version 1.1.
-% This license is in the accompanying file OFL.txt, and is also
-% available with a FAQ at: http://scripts.sil.org/OFL.
-%%EndComments
-FontDirectory/CMSY8 known{/CMSY8 findfont dup/UniqueID known{dup
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-{save true}{false}ifelse}{false}ifelse
-11 dict begin
-/FontType 1 def
-/FontMatrix [0.001 0 0 0.001 0 0 ]readonly def
-/FontName /CMSY8 def
-/FontBBox {-30 -955 1185 779 }readonly def
-/UniqueID 5096649 def
-/PaintType 0 def
-/FontInfo 9 dict dup begin
-/version (003.002) readonly def
-/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMSY8.) readonly def
-/FullName (CMSY8) readonly def
-/FamilyName (Computer Modern) readonly def
-/Weight (Medium) readonly def
-/ItalicAngle -14.04 def
-/isFixedPitch false def
-/UnderlinePosition -100 def
-/UnderlineThickness 50 def
-end readonly def
-/Encoding 256 array
-0 1 255 {1 index exch /.notdef put} for
-dup 0 /minus put
-dup 33 /arrowright put
-dup 49 /infinity put
-dup 112 /radical put
-readonly def
-currentdict end
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+%%BeginFont: CMSY8
+%!PS-AdobeFont-1.0: CMSY8 003.002
+%%Title: CMSY8
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMSY8.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% available with a FAQ at: http://scripts.sil.org/OFL.
+%%EndComments
+FontDirectory/CMSY8 known{/CMSY8 findfont dup/UniqueID known{dup
+/UniqueID get 5096649 eq exch/FontType get 1 eq and}{pop false}ifelse
+{save true}{false}ifelse}{false}ifelse
+11 dict begin
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0 ]readonly def
+/FontName /CMSY8 def
+/FontBBox {-30 -955 1185 779 }readonly def
+/UniqueID 5096649 def
+/PaintType 0 def
+/FontInfo 9 dict dup begin
+/version (003.002) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMSY8.) readonly def
+/FullName (CMSY8) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
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+/isFixedPitch false def
+/UnderlinePosition -100 def
+/UnderlineThickness 50 def
+end readonly def
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+dup 102 /braceleft put
+dup 103 /braceright put
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+%%BeginFont: CMMI8
+%!PS-AdobeFont-1.0: CMMI8 003.002
+%%Title: CMMI8
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMMI8.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% available with a FAQ at: http://scripts.sil.org/OFL.
+%%EndComments
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+/FontName /CMMI8 def
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+/version (003.002) readonly def
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+/FamilyName (Computer Modern) readonly def
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+%%EndFont 
+%%BeginFont: CMSY10
+%!PS-AdobeFont-1.0: CMSY10 003.002
+%%Title: CMSY10
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMSY10.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% available with a FAQ at: http://scripts.sil.org/OFL.
+%%EndComments
+FontDirectory/CMSY10 known{/CMSY10 findfont dup/UniqueID known{dup
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+{save true}{false}ifelse}{false}ifelse
+11 dict begin
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+/FontName /CMSY10 def
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+/version (003.002) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMSY10.) readonly def
+/FullName (CMSY10) readonly def
+/FamilyName (Computer Modern) readonly def
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+/isFixedPitch false def
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+/UnderlineThickness 50 def
+end readonly def
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+dup 54 /negationslash put
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+{restore}if
+%%EndFont 
+%%BeginFont: CMR8
+%!PS-AdobeFont-1.0: CMR8 003.002
+%%Title: CMR8
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMR8.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% available with a FAQ at: http://scripts.sil.org/OFL.
+%%EndComments
+FontDirectory/CMR8 known{/CMR8 findfont dup/UniqueID known{dup
+/UniqueID get 5000791 eq exch/FontType get 1 eq and}{pop false}ifelse
+{save true}{false}ifelse}{false}ifelse
+11 dict begin
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0 ]readonly def
+/FontName /CMR8 def
+/FontBBox {-36 -250 1070 750 }readonly def
+/UniqueID 5000791 def
+/PaintType 0 def
+/FontInfo 9 dict dup begin
+/version (003.002) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMR8.) readonly def
+/FullName (CMR8) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
+/ItalicAngle 0 def
+/isFixedPitch false def
+/UnderlinePosition -100 def
+/UnderlineThickness 50 def
+end readonly def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 33 /exclam put
+dup 34 /quotedblright put
+dup 37 /percent put
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+dup 41 /parenright put
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+dup 91 /bracketleft put
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+dup 114 /r put
+dup 115 /s put
+dup 116 /t put
+dup 117 /u put
+dup 118 /v put
+dup 120 /x put
+dup 121 /y put
+readonly def
+currentdict end
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@@ -5927,32 +5683,32 @@ CF1CC8D4FDF229B40B7077483EB50B1117D644D807A94FCC82F502FD4C8D6C4D
 cleartomark
 {restore}if
 %%EndFont 
-%%BeginFont: CMMI8
-%!PS-AdobeFont-1.0: CMMI8 003.002
-%%Title: CMMI8
+%%BeginFont: CMMI10
+%!PS-AdobeFont-1.0: CMMI10 003.002
+%%Title: CMMI10
 %Version: 003.002
 %%CreationDate: Mon Jul 13 16:17:00 2009
 %%Creator: David M. Jones
 %Copyright: Copyright (c) 1997, 2009 American Mathematical Society
-%Copyright: (<http://www.ams.org>), with Reserved Font Name CMMI8.
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMMI10.
 % This Font Software is licensed under the SIL Open Font License, Version 1.1.
 % This license is in the accompanying file OFL.txt, and is also
 % available with a FAQ at: http://scripts.sil.org/OFL.
 %%EndComments
-FontDirectory/CMMI8 known{/CMMI8 findfont dup/UniqueID known{dup
-/UniqueID get 5087383 eq exch/FontType get 1 eq and}{pop false}ifelse
+FontDirectory/CMMI10 known{/CMMI10 findfont dup/UniqueID known{dup
+/UniqueID get 5087385 eq exch/FontType get 1 eq and}{pop false}ifelse
 {save true}{false}ifelse}{false}ifelse
 11 dict begin
 /FontType 1 def
 /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def
-/FontName /CMMI8 def
-/FontBBox {-24 -250 1110 750 }readonly def
-/UniqueID 5087383 def
+/FontName /CMMI10 def
+/FontBBox {-32 -250 1048 750 }readonly def
+/UniqueID 5087385 def
 /PaintType 0 def
 /FontInfo 10 dict dup begin
 /version (003.002) readonly def
-/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMMI8.) readonly def
-/FullName (CMMI8) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMMI10.) readonly def
+/FullName (CMMI10) readonly def
 /FamilyName (Computer Modern) readonly def
 /Weight (Medium) readonly def
 /ItalicAngle -14.04 def
@@ -5963,294 +5719,395 @@ FontDirectory/CMMI8 known{/CMMI8 findfont dup/UniqueID known{dup
 end readonly def
 /Encoding 256 array
 0 1 255 {1 index exch /.notdef put} for
-dup 11 /alpha put
+dup 22 /mu put
+dup 25 /pi put
+dup 58 /period put
 dup 59 /comma put
+dup 60 /less put
 dup 61 /slash put
+dup 67 /C put
+dup 69 /E put
 dup 78 /N put
+dup 80 /P put
 dup 83 /S put
+dup 84 /T put
 dup 88 /X put
+dup 90 /Z put
+dup 97 /a put
+dup 100 /d put
 dup 105 /i put
 dup 106 /j put
 dup 107 /k put
+dup 109 /m put
 dup 110 /n put
-dup 111 /o put
 dup 112 /p put
 dup 113 /q put
-readonly def
-currentdict end
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 0000000000000000000000000000000000000000000000000000000000000000
@@ -6262,32 +6119,32 @@ B61BAEA85FF807D174B8DEF85FD9DFDF0A8884A77B53792784270115D43B32E9
 cleartomark
 {restore}if
 %%EndFont 
-%%BeginFont: CMR8
-%!PS-AdobeFont-1.0: CMR8 003.002
-%%Title: CMR8
+%%BeginFont: CMR10
+%!PS-AdobeFont-1.0: CMR10 003.002
+%%Title: CMR10
 %Version: 003.002
 %%CreationDate: Mon Jul 13 16:17:00 2009
 %%Creator: David M. Jones
 %Copyright: Copyright (c) 1997, 2009 American Mathematical Society
-%Copyright: (<http://www.ams.org>), with Reserved Font Name CMR8.
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMR10.
 % This Font Software is licensed under the SIL Open Font License, Version 1.1.
 % This license is in the accompanying file OFL.txt, and is also
 % available with a FAQ at: http://scripts.sil.org/OFL.
 %%EndComments
-FontDirectory/CMR8 known{/CMR8 findfont dup/UniqueID known{dup
-/UniqueID get 5000791 eq exch/FontType get 1 eq and}{pop false}ifelse
+FontDirectory/CMR10 known{/CMR10 findfont dup/UniqueID known{dup
+/UniqueID get 5000793 eq exch/FontType get 1 eq and}{pop false}ifelse
 {save true}{false}ifelse}{false}ifelse
 11 dict begin
 /FontType 1 def
 /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def
-/FontName /CMR8 def
-/FontBBox {-36 -250 1070 750 }readonly def
-/UniqueID 5000791 def
+/FontName /CMR10 def
+/FontBBox {-40 -250 1009 750 }readonly def
+/UniqueID 5000793 def
 /PaintType 0 def
 /FontInfo 9 dict dup begin
 /version (003.002) readonly def
-/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMR8.) readonly def
-/FullName (CMR8) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMR10.) readonly def
+/FullName (CMR10) readonly def
 /FamilyName (Computer Modern) readonly def
 /Weight (Medium) readonly def
 /ItalicAngle 0 def
@@ -6297,9 +6154,18 @@ FontDirectory/CMR8 known{/CMR8 findfont dup/UniqueID known{dup
 end readonly def
 /Encoding 256 array
 0 1 255 {1 index exch /.notdef put} for
+dup 11 /ff put
+dup 12 /fi put
+dup 22 /macron put
+dup 34 /quotedblright put
+dup 37 /percent put
+dup 39 /quoteright put
 dup 40 /parenleft put
 dup 41 /parenright put
 dup 43 /plus put
+dup 44 /comma put
+dup 45 /hyphen put
+dup 46 /period put
 dup 48 /zero put
 dup 49 /one put
 dup 50 /two put
@@ -6310,8 +6176,51 @@ dup 54 /six put
 dup 55 /seven put
 dup 56 /eight put
 dup 57 /nine put
+dup 58 /colon put
 dup 61 /equal put
+dup 65 /A put
+dup 66 /B put
+dup 67 /C put
+dup 70 /F put
+dup 71 /G put
+dup 72 /H put
+dup 73 /I put
+dup 76 /L put
+dup 77 /M put
+dup 78 /N put
+dup 79 /O put
+dup 80 /P put
+dup 83 /S put
+dup 84 /T put
+dup 85 /U put
+dup 87 /W put
 dup 94 /circumflex put
+dup 97 /a put
+dup 98 /b put
+dup 99 /c put
+dup 100 /d put
+dup 101 /e put
+dup 102 /f put
+dup 103 /g put
+dup 104 /h put
+dup 105 /i put
+dup 106 /j put
+dup 107 /k put
+dup 108 /l put
+dup 109 /m put
+dup 110 /n put
+dup 111 /o put
+dup 112 /p put
+dup 113 /q put
+dup 114 /r put
+dup 115 /s put
+dup 116 /t put
+dup 117 /u put
+dup 118 /v put
+dup 119 /w put
+dup 120 /x put
+dup 121 /y put
+dup 122 /z put
 readonly def
 currentdict end
 currentfile eexec
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+%%BeginFont: CMBX10
+%!PS-AdobeFont-1.0: CMBX10 003.002
+%%Title: CMBX10
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMBX10.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% available with a FAQ at: http://scripts.sil.org/OFL.
+%%EndComments
+FontDirectory/CMBX10 known{/CMBX10 findfont dup/UniqueID known{dup
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+/FontName /CMBX10 def
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+/version (003.002) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMBX10.) readonly def
+/FullName (CMBX10) readonly def
+/FamilyName (Computer Modern) readonly def
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+end readonly def
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 %%Trailer
 
 userdict /end-hook known{end-hook}if

diff --git a/rapport.tex b/rapport.tex
index 5bf2b8b82416608b451b7e6fd3a7c6f30fc47a48..95cf5e49fa3158de84034eadb9d44e5d46afd6df 100644 (file)
--- a/rapport.tex
+++ b/rapport.tex
@@ -12,7 +12,7 @@
 \setcounter{secnumdepth}{0}
 
 \lstset{ %
-  basicstyle=\footnotesize,       % the size of the fonts
+  basicstyle=\scriptsize,       % the size of the fonts
   %numbers=left,                   % where to put the line-numbers
   %numberstyle=\footnotesize,      % the size of the fonts that are used for the
   %line-numbers
@@ -235,7 +235,7 @@ conditioning on the outcome of the next jump, we get
   \label{eq:mui}
   \begin{split}
   \mu_i &= p + q(1 + \mu_{i-1}+\mu_{i}) \text{ for } i = 1,...,N-1, \\
-  p\mu_i &= \frac{1}{p} +\frac{q}{p}\mu_{i-1}.
+  p\mu_i &= \frac{1}{p} + \frac{q}{p}\mu_{i-1}.
   \end{split}
 \end{equation}
 When the Markov chain is in state 0, going to state 1 follows a
@@ -260,7 +260,7 @@ The expected time of going from state i to state n is then given by
   \begin{split}
   E\left(N_{i,n}\right) &= \sum_{j = i}^{n-1}\mu_j\\
   &= \sum_{j=i}^{n-1}\left(\frac{1}{p_0}+\frac{j}{p}\right) \\
-  &= (n-i)(\frac{1}{p_0} + \frac{i+n-1}{2p})
+  &= (n-i)\left(\frac{1}{p_0} + \frac{i+n-1}{2p}\right)
   \end{split}
 \end{equation}
 This gives $E(N_{25,100})=9450$, $E(N_{50,100})=7550$ and $E(N_{75,100})=4400$.
@@ -289,7 +289,7 @@ $r_0 = 1$, making it possible to get stuck in state $0$ and $100$, and the
 estimated probability of ending up in state $100$ is shown in table
 \ref{tab:ratio_to_100}.
 
-\begin{table}
+\begin{table}[h]
   \caption{\sf Avg. number of jumps before ending up in state $100$, over $5000$ runs}
   \label{tab:jumps_to_100}
   \begin{center}
@@ -439,7 +439,8 @@ spends in the different states depending on the initial state. We first look at
 submatrix for the class $\{0,1\}$, that is
 \begin{equation}
   \label{mat:small}
-  \bordermatrix{~ & 0 & 1 \cr 0 & 1/2 & 1/2 \cr 1 & 1/3 & 2/3}. 
+  \bordermatrix{~ & 0 & 1 \cr 0 & \sfrac{1}{2} & \sfrac{1}{2} \cr 1 &
+\sfrac{1}{3} & \sfrac{2}{3}}. 
 \end{equation}
 In a recurrent class as this the long run proportion of time
 spent in either state is independent of the initial state. We find the
@@ -498,12 +499,12 @@ time spent in a state given an initial state $i$, $m_{i,j}$, is as follows
 \begin{equation}
   (m_{i,j}) = 
   \begin{bmatrix}
-  \frac{4}{10} & \frac{6}{10} & 0  & 0 & 0 & 0 \\
-  \frac{4}{10} & \frac{6}{10} & 0  & 0 & 0 & 0 \\
-  \frac{16}{85} & \frac{24}{85} & 0 & 0 & \frac{9}{34} & \frac{9}{34} \\
-  \frac{14}{85} & \frac{21}{85} & 0 & 0 & \frac{10}{34} & \frac{10}{34} \\
-  0 & 0 & 0  & 0 & \frac{1}{2} & \frac{1}{2} \\
-  0 & 0 & 0  & 0 & \frac{1}{2} & \frac{1}{2} \\
+  \sfrac{4}{10} & \sfrac{6}{10} & 0  & 0 & 0 & 0 \\
+  \sfrac{4}{10} & \sfrac{6}{10} & 0  & 0 & 0 & 0 \\
+  \sfrac{16}{85} & \sfrac{24}{85} & 0 & 0 & \sfrac{9}{34} & \sfrac{9}{34} \\
+  \sfrac{14}{85} & \sfrac{21}{85} & 0 & 0 & \sfrac{10}{34} & \sfrac{10}{34} \\
+  0 & 0 & 0  & 0 & \sfrac{1}{2} & \sfrac{1}{2} \\
+  0 & 0 & 0  & 0 & \sfrac{1}{2} & \sfrac{1}{2} \\
   \end{bmatrix}
 \end{equation}