From: Bjørn Rustad Date: Fri, 28 Oct 2011 15:03:53 +0000 (+0200) Subject: Ferdig for dagen X-Git-Url: http://git.rustad.me/?a=commitdiff_plain;h=3d0d1670b10b45347d6d9ebb940d29d9f5ef622d;p=stokpros1 Ferdig for dagen --- diff --git a/rapport.aux b/rapport.aux index 5f0960f..4951813 100644 --- a/rapport.aux +++ b/rapport.aux @@ -27,31 +27,31 @@ \@writefile{toc}{\contentsline {subsection}{c)}{5}} \@writefile{toc}{\contentsline {subsection}{d)}{5}} \@writefile{lol}{\contentsline {lstlisting}{simulate.py}{5}} -\@writefile{toc}{\contentsline {subsubsection}{Confidence intervals}{6}} -\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces \sf Avg. number of jumps before ending up in state $100$, over $5000$ runs}}{7}} -\newlabel{tab:jumps_to_100}{{1}{7}} -\@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 index 842a874..3273cdd 100644 Binary files a/rapport.dvi and b/rapport.dvi differ diff --git a/rapport.log b/rapport.log index b290a4a..476cc68 100644 --- a/rapport.log +++ b/rapport.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.1415926-2.3-1.40.12 (TeX Live 2011) (format=latex 2011.8.11) 28 OCT 2011 16:27 +This is pdfTeX, Version 3.1415926-2.3-1.40.12 (TeX Live 2011) (format=latex 2011.8.11) 28 OCT 2011 16:55 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -814,54 +814,40 @@ LaTeX Font Info: Try loading font information for OML+cmr on input line 15. 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{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: (), with Reserved Font Name CMR10. +%Copyright: (), 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\051, with Reserved Font Name CMR10.) readonly def -/FullName (CMR10) readonly def +/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\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 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Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society -%Copyright: (), with Reserved Font Name CMBX10. +%Copyright: (), 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 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Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society -%Copyright: (), with Reserved Font Name CMR8. +%Copyright: (), 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\051, with Reserved Font Name CMR8.) readonly def -/FullName (CMR8) readonly def +/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\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 @@ -6321,251 +6230,973 @@ D9D66F633B846AB284BCF8B0411B772DE5CE3DD325E55798292D7BD972BD75FA 7C56816B03B901BF4A25A07175380E50A213F877C44778B3C5AADBCC86D6E551 E6AF364B0BFCAAD22D8D558C5C81A7D425A1629DD5182206742D1D082A12F078 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Fp(\031)25 b Fs(0)p Fr(:)p Fs(263)p +Fr(:)3090 3920 y Fs(\(38\))501 4190 y(These)e(results)f(all)i(coincide) +g(w)m(ell)f(with)g(the)g(results)f(from)h Fg(c\))p Fs(,)h(thereb)m(y)f +(making)501 4303 y(them)31 b(b)s(eliev)-5 b(able.)p Black +1828 5637 a Fm(11)p Black eop end %%Trailer userdict /end-hook known{end-hook}if diff --git a/rapport.tex b/rapport.tex index 5bf2b8b..95cf5e4 100644 --- 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}