This is pdfTeX, Version 3.141592653-2.6-1.40.27 (TeX Live 2026/dev/Arch Linux) (preloaded format=pdflatex 2025.11.1) 21 NOV 2025 18:59 entering extended mode restricted \write18 enabled. %&-line parsing enabled. **"Probability 564.tex" (./Probability 564.tex LaTeX2e <2024-11-01> patch level 2 L3 programming layer <2025-01-18> (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2024/06/29 v1.4n Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size10.clo File: size10.clo 2024/06/29 v1.4n Standard LaTeX file (size option) ) \c@part=\count196 \c@section=\count197 \c@subsection=\count198 \c@subsubsection=\count199 \c@paragraph=\count266 \c@subparagraph=\count267 \c@figure=\count268 \c@table=\count269 \abovecaptionskip=\skip49 \belowcaptionskip=\skip50 \bibindent=\dimen141 ) (/usr/share/texmf-dist/tex/latex/amsmath/amsmath.sty Package: amsmath 2024/11/05 v2.17t AMS math features \@mathmargin=\skip51 For additional information on amsmath, use the `?' option. (/usr/share/texmf-dist/tex/latex/amsmath/amstext.sty Package: amstext 2021/08/26 v2.01 AMS text (/usr/share/texmf-dist/tex/latex/amsmath/amsgen.sty File: amsgen.sty 1999/11/30 v2.0 generic functions \@emptytoks=\toks17 \ex@=\dimen142 )) (/usr/share/texmf-dist/tex/latex/amsmath/amsbsy.sty Package: amsbsy 1999/11/29 v1.2d Bold Symbols \pmbraise@=\dimen143 ) (/usr/share/texmf-dist/tex/latex/amsmath/amsopn.sty Package: amsopn 2022/04/08 v2.04 operator names ) \inf@bad=\count270 LaTeX Info: Redefining \frac on input line 233. \uproot@=\count271 \leftroot@=\count272 LaTeX Info: Redefining \overline on input line 398. LaTeX Info: Redefining \colon on input line 409. \classnum@=\count273 \DOTSCASE@=\count274 LaTeX Info: Redefining \ldots on input line 495. LaTeX Info: Redefining \dots on input line 498. LaTeX Info: Redefining \cdots on input line 619. \Mathstrutbox@=\box52 \strutbox@=\box53 LaTeX Info: Redefining \big on input line 721. LaTeX Info: Redefining \Big on input line 722. LaTeX Info: Redefining \bigg on input line 723. LaTeX Info: Redefining \Bigg on input line 724. \big@size=\dimen144 LaTeX Font Info: Redeclaring font encoding OML on input line 742. LaTeX Font Info: Redeclaring font encoding OMS on input line 743. \macc@depth=\count275 LaTeX Info: Redefining \bmod on input line 904. LaTeX Info: Redefining \pmod on input line 909. LaTeX Info: Redefining \smash on input line 939. LaTeX Info: Redefining \relbar on input line 969. LaTeX Info: Redefining \Relbar on input line 970. \c@MaxMatrixCols=\count276 \dotsspace@=\muskip17 \c@parentequation=\count277 \dspbrk@lvl=\count278 \tag@help=\toks18 \row@=\count279 \column@=\count280 \maxfields@=\count281 \andhelp@=\toks19 \eqnshift@=\dimen145 \alignsep@=\dimen146 \tagshift@=\dimen147 \tagwidth@=\dimen148 \totwidth@=\dimen149 \lineht@=\dimen150 \@envbody=\toks20 \multlinegap=\skip52 \multlinetaggap=\skip53 \mathdisplay@stack=\toks21 LaTeX Info: Redefining \[ on input line 2953. LaTeX Info: Redefining \] on input line 2954. ) (/usr/share/texmf-dist/tex/latex/amscls/amsthm.sty Package: amsthm 2020/05/29 v2.20.6 \thm@style=\toks22 \thm@bodyfont=\toks23 \thm@headfont=\toks24 \thm@notefont=\toks25 \thm@headpunct=\toks26 \thm@preskip=\skip54 \thm@postskip=\skip55 \thm@headsep=\skip56 \dth@everypar=\toks27 ) (/usr/share/texmf-dist/tex/latex/amsfonts/amssymb.sty Package: amssymb 2013/01/14 v3.01 AMS font symbols (/usr/share/texmf-dist/tex/latex/amsfonts/amsfonts.sty Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support \symAMSa=\mathgroup4 \symAMSb=\mathgroup5 LaTeX Font Info: Redeclaring math symbol \hbar on input line 98. LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' (Font) U/euf/m/n --> U/euf/b/n on input line 106. )) (/usr/share/texmf-dist/tex/latex/mathtools/mathtools.sty Package: mathtools 2024/10/04 v1.31 mathematical typesetting tools (/usr/share/texmf-dist/tex/latex/graphics/keyval.sty Package: keyval 2022/05/29 v1.15 key=value parser (DPC) \KV@toks@=\toks28 ) (/usr/share/texmf-dist/tex/latex/tools/calc.sty Package: calc 2023/07/08 v4.3 Infix arithmetic (KKT,FJ) \calc@Acount=\count282 \calc@Bcount=\count283 \calc@Adimen=\dimen151 \calc@Bdimen=\dimen152 \calc@Askip=\skip57 \calc@Bskip=\skip58 LaTeX Info: Redefining \setlength on input line 80. LaTeX Info: Redefining \addtolength on input line 81. \calc@Ccount=\count284 \calc@Cskip=\skip59 ) (/usr/share/texmf-dist/tex/latex/mathtools/mhsetup.sty Package: mhsetup 2021/03/18 v1.4 programming setup (MH) ) \g_MT_multlinerow_int=\count285 \l_MT_multwidth_dim=\dimen153 \origjot=\skip60 \l_MT_shortvdotswithinadjustabove_dim=\dimen154 \l_MT_shortvdotswithinadjustbelow_dim=\dimen155 \l_MT_above_intertext_sep=\dimen156 \l_MT_below_intertext_sep=\dimen157 \l_MT_above_shortintertext_sep=\dimen158 \l_MT_below_shortintertext_sep=\dimen159 \xmathstrut@box=\box54 \xmathstrut@dim=\dimen160 ) (/usr/share/texmf-dist/tex/latex/bbm-macros/bbm.sty Package: bbm 1999/03/15 V 1.2 provides fonts for set symbols - TH LaTeX Font Info: Overwriting math alphabet `\mathbbm' in version `bold' (Font) U/bbm/m/n --> U/bbm/bx/n on input line 33. LaTeX Font Info: Overwriting math alphabet `\mathbbmss' in version `bold' (Font) U/bbmss/m/n --> U/bbmss/bx/n on input line 35. ) (/usr/share/texmf-dist/tex/latex/jknapltx/mathrsfs.sty Package: mathrsfs 1996/01/01 Math RSFS package v1.0 (jk) \symrsfs=\mathgroup6 ) (/usr/share/texmf-dist/tex/latex/tikz-cd/tikz-cd.sty Package: tikz-cd 2021/05/04 v1.0 Commutative diagrams with TikZ (/usr/share/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty (/usr/share/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty (/usr/share/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty (/usr/share/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex \pgfutil@everybye=\toks29 \pgfutil@tempdima=\dimen161 \pgfutil@tempdimb=\dimen162 ) (/usr/share/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def \pgfutil@abb=\box55 ) (/usr/share/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex (/usr/share/texmf-dist/tex/generic/pgf/pgf.revision.tex) Package: pgfrcs 2023-01-15 v3.1.10 (3.1.10) )) Package: pgf 2023-01-15 v3.1.10 (3.1.10) (/usr/share/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty (/usr/share/texmf-dist/tex/latex/graphics/graphicx.sty Package: graphicx 2021/09/16 v1.2d Enhanced LaTeX Graphics (DPC,SPQR) (/usr/share/texmf-dist/tex/latex/graphics/graphics.sty Package: graphics 2024/08/06 v1.4g Standard LaTeX Graphics (DPC,SPQR) (/usr/share/texmf-dist/tex/latex/graphics/trig.sty Package: trig 2023/12/02 v1.11 sin cos tan (DPC) ) (/usr/share/texmf-dist/tex/latex/graphics-cfg/graphics.cfg File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration ) Package graphics Info: Driver file: pdftex.def on input line 106. (/usr/share/texmf-dist/tex/latex/graphics-def/pdftex.def File: pdftex.def 2024/04/13 v1.2c Graphics/color driver for pdftex )) \Gin@req@height=\dimen163 \Gin@req@width=\dimen164 ) (/usr/share/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty (/usr/share/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex Package: pgfsys 2023-01-15 v3.1.10 (3.1.10) (/usr/share/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex \pgfkeys@pathtoks=\toks30 \pgfkeys@temptoks=\toks31 (/usr/share/texmf-dist/tex/generic/pgf/utilities/pgfkeyslibraryfiltered.code.te x \pgfkeys@tmptoks=\toks32 )) \pgf@x=\dimen165 \pgf@y=\dimen166 \pgf@xa=\dimen167 \pgf@ya=\dimen168 \pgf@xb=\dimen169 \pgf@yb=\dimen170 \pgf@xc=\dimen171 \pgf@yc=\dimen172 \pgf@xd=\dimen173 \pgf@yd=\dimen174 \w@pgf@writea=\write3 \r@pgf@reada=\read2 \c@pgf@counta=\count286 \c@pgf@countb=\count287 \c@pgf@countc=\count288 \c@pgf@countd=\count289 \t@pgf@toka=\toks33 \t@pgf@tokb=\toks34 \t@pgf@tokc=\toks35 \pgf@sys@id@count=\count290 (/usr/share/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg File: pgf.cfg 2023-01-15 v3.1.10 (3.1.10) ) Driver file for pgf: pgfsys-pdftex.def (/usr/share/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def File: pgfsys-pdftex.def 2023-01-15 v3.1.10 (3.1.10) (/usr/share/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def File: pgfsys-common-pdf.def 2023-01-15 v3.1.10 (3.1.10) ))) (/usr/share/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex File: pgfsyssoftpath.code.tex 2023-01-15 v3.1.10 (3.1.10) \pgfsyssoftpath@smallbuffer@items=\count291 \pgfsyssoftpath@bigbuffer@items=\count292 ) (/usr/share/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex File: pgfsysprotocol.code.tex 2023-01-15 v3.1.10 (3.1.10) )) (/usr/share/texmf-dist/tex/latex/xcolor/xcolor.sty Package: xcolor 2024/09/29 v3.02 LaTeX color extensions (UK) (/usr/share/texmf-dist/tex/latex/graphics-cfg/color.cfg File: color.cfg 2016/01/02 v1.6 sample color configuration ) Package xcolor Info: Driver file: pdftex.def on input line 274. (/usr/share/texmf-dist/tex/latex/graphics/mathcolor.ltx) Package xcolor Info: Model `cmy' substituted by `cmy0' on input line 1349. Package xcolor Info: Model `hsb' substituted by `rgb' on input line 1353. Package xcolor Info: Model `RGB' extended on input line 1365. Package xcolor Info: Model `HTML' substituted by `rgb' on input line 1367. Package xcolor Info: Model `Hsb' substituted by `hsb' on input line 1368. Package xcolor Info: Model `tHsb' substituted by `hsb' on input line 1369. Package xcolor Info: Model `HSB' substituted by `hsb' on input line 1370. Package xcolor Info: Model `Gray' substituted by `gray' on input line 1371. Package xcolor Info: Model `wave' substituted by `hsb' on input line 1372. ) (/usr/share/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex Package: pgfcore 2023-01-15 v3.1.10 (3.1.10) (/usr/share/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex (/usr/share/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex) (/usr/share/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex \pgfmath@dimen=\dimen175 \pgfmath@count=\count293 \pgfmath@box=\box56 \pgfmath@toks=\toks36 \pgfmath@stack@operand=\toks37 \pgfmath@stack@operation=\toks38 ) (/usr/share/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex) (/usr/share/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex) (/usr/share/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code .tex) (/usr/share/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex) (/usr/share/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.code.te x) (/usr/share/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex) (/usr/share/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex) (/usr/share/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex) (/usr/share/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics .code.tex) (/usr/share/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex) (/usr/share/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex \c@pgfmathroundto@lastzeros=\count294 )) (/usr/share/texmf-dist/tex/generic/pgf/math/pgfint.code.tex) (/usr/share/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex File: pgfcorepoints.code.tex 2023-01-15 v3.1.10 (3.1.10) \pgf@picminx=\dimen176 \pgf@picmaxx=\dimen177 \pgf@picminy=\dimen178 \pgf@picmaxy=\dimen179 \pgf@pathminx=\dimen180 \pgf@pathmaxx=\dimen181 \pgf@pathminy=\dimen182 \pgf@pathmaxy=\dimen183 \pgf@xx=\dimen184 \pgf@xy=\dimen185 \pgf@yx=\dimen186 \pgf@yy=\dimen187 \pgf@zx=\dimen188 \pgf@zy=\dimen189 ) (/usr/share/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex File: pgfcorepathconstruct.code.tex 2023-01-15 v3.1.10 (3.1.10) \pgf@path@lastx=\dimen190 \pgf@path@lasty=\dimen191 ) (/usr/share/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex File: pgfcorepathusage.code.tex 2023-01-15 v3.1.10 (3.1.10) \pgf@shorten@end@additional=\dimen192 \pgf@shorten@start@additional=\dimen193 ) (/usr/share/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex File: pgfcorescopes.code.tex 2023-01-15 v3.1.10 (3.1.10) \pgfpic=\box57 \pgf@hbox=\box58 \pgf@layerbox@main=\box59 \pgf@picture@serial@count=\count295 ) (/usr/share/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex File: pgfcoregraphicstate.code.tex 2023-01-15 v3.1.10 (3.1.10) \pgflinewidth=\dimen194 ) (/usr/share/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.code.t ex File: pgfcoretransformations.code.tex 2023-01-15 v3.1.10 (3.1.10) \pgf@pt@x=\dimen195 \pgf@pt@y=\dimen196 \pgf@pt@temp=\dimen197 ) (/usr/share/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex File: pgfcorequick.code.tex 2023-01-15 v3.1.10 (3.1.10) ) (/usr/share/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex File: pgfcoreobjects.code.tex 2023-01-15 v3.1.10 (3.1.10) ) (/usr/share/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.te x File: pgfcorepathprocessing.code.tex 2023-01-15 v3.1.10 (3.1.10) ) (/usr/share/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex File: pgfcorearrows.code.tex 2023-01-15 v3.1.10 (3.1.10) \pgfarrowsep=\dimen198 ) (/usr/share/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex File: pgfcoreshade.code.tex 2023-01-15 v3.1.10 (3.1.10) \pgf@max=\dimen199 \pgf@sys@shading@range@num=\count296 \pgf@shadingcount=\count297 ) (/usr/share/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex File: pgfcoreimage.code.tex 2023-01-15 v3.1.10 (3.1.10) ) (/usr/share/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex File: pgfcoreexternal.code.tex 2023-01-15 v3.1.10 (3.1.10) \pgfexternal@startupbox=\box60 ) (/usr/share/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex File: pgfcorelayers.code.tex 2023-01-15 v3.1.10 (3.1.10) ) (/usr/share/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex File: pgfcoretransparency.code.tex 2023-01-15 v3.1.10 (3.1.10) ) (/usr/share/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex File: pgfcorepatterns.code.tex 2023-01-15 v3.1.10 (3.1.10) ) (/usr/share/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex File: pgfcorerdf.code.tex 2023-01-15 v3.1.10 (3.1.10) ))) (/usr/share/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.tex File: pgfmoduleshapes.code.tex 2023-01-15 v3.1.10 (3.1.10) \pgfnodeparttextbox=\box61 ) (/usr/share/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex File: pgfmoduleplot.code.tex 2023-01-15 v3.1.10 (3.1.10) ) (/usr/share/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty Package: pgfcomp-version-0-65 2023-01-15 v3.1.10 (3.1.10) \pgf@nodesepstart=\dimen256 \pgf@nodesepend=\dimen257 ) (/usr/share/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty Package: pgfcomp-version-1-18 2023-01-15 v3.1.10 (3.1.10) )) (/usr/share/texmf-dist/tex/latex/pgf/utilities/pgffor.sty (/usr/share/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty (/usr/share/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex)) (/usr/share/texmf-dist/tex/latex/pgf/math/pgfmath.sty (/usr/share/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex)) (/usr/share/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex Package: pgffor 2023-01-15 v3.1.10 (3.1.10) \pgffor@iter=\dimen258 \pgffor@skip=\dimen259 \pgffor@stack=\toks39 \pgffor@toks=\toks40 )) (/usr/share/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex Package: tikz 2023-01-15 v3.1.10 (3.1.10) (/usr/share/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.te x File: pgflibraryplothandlers.code.tex 2023-01-15 v3.1.10 (3.1.10) \pgf@plot@mark@count=\count298 \pgfplotmarksize=\dimen260 ) \tikz@lastx=\dimen261 \tikz@lasty=\dimen262 \tikz@lastxsaved=\dimen263 \tikz@lastysaved=\dimen264 \tikz@lastmovetox=\dimen265 \tikz@lastmovetoy=\dimen266 \tikzleveldistance=\dimen267 \tikzsiblingdistance=\dimen268 \tikz@figbox=\box62 \tikz@figbox@bg=\box63 \tikz@tempbox=\box64 \tikz@tempbox@bg=\box65 \tikztreelevel=\count299 \tikznumberofchildren=\count300 \tikznumberofcurrentchild=\count301 \tikz@fig@count=\count302 (/usr/share/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex File: pgfmodulematrix.code.tex 2023-01-15 v3.1.10 (3.1.10) \pgfmatrixcurrentrow=\count303 \pgfmatrixcurrentcolumn=\count304 \pgf@matrix@numberofcolumns=\count305 ) \tikz@expandcount=\count306 (/usr/share/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrary topaths.code.tex File: tikzlibrarytopaths.code.tex 2023-01-15 v3.1.10 (3.1.10) ))) (/usr/share/texmf-dist/tex/generic/tikz-cd/tikzlibrarycd.code.tex (/usr/share/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrary matrix.code.tex File: tikzlibrarymatrix.code.tex 2023-01-15 v3.1.10 (3.1.10) ) (/usr/share/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrary quotes.code.tex File: tikzlibraryquotes.code.tex 2023-01-15 v3.1.10 (3.1.10) ) (/usr/share/texmf-dist/tex/generic/pgf/libraries/pgflibraryarrows.meta.code.tex File: pgflibraryarrows.meta.code.tex 2023-01-15 v3.1.10 (3.1.10) \pgfarrowinset=\dimen269 \pgfarrowlength=\dimen270 \pgfarrowwidth=\dimen271 \pgfarrowlinewidth=\dimen272 ))) (/usr/share/texmf-dist/tex/latex/geometry/geometry.sty Package: geometry 2020/01/02 v5.9 Page Geometry (/usr/share/texmf-dist/tex/generic/iftex/ifvtex.sty Package: ifvtex 2019/10/25 v1.7 ifvtex legacy package. Use iftex instead. (/usr/share/texmf-dist/tex/generic/iftex/iftex.sty Package: iftex 2024/12/12 v1.0g TeX engine tests )) \Gm@cnth=\count307 \Gm@cntv=\count308 \c@Gm@tempcnt=\count309 \Gm@bindingoffset=\dimen273 \Gm@wd@mp=\dimen274 \Gm@odd@mp=\dimen275 \Gm@even@mp=\dimen276 \Gm@layoutwidth=\dimen277 \Gm@layoutheight=\dimen278 \Gm@layouthoffset=\dimen279 \Gm@layoutvoffset=\dimen280 \Gm@dimlist=\toks41 ) \c@problem=\count310 \c@solution=\count311 \c@example=\count312 \c@definition=\count313 \c@proposition=\count314 \c@theorem=\count315 (/usr/share/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def File: l3backend-pdftex.def 2024-05-08 L3 backend support: PDF output (pdfTeX) \l__color_backend_stack_int=\count316 \l__pdf_internal_box=\box66 ) (./Probability 564.aux) \openout1 = `"Probability 564.aux"'. LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 32. LaTeX Font Info: ... okay on input line 32. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 32. LaTeX Font Info: ... okay on input line 32. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 32. LaTeX Font Info: ... okay on input line 32. LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 32. LaTeX Font Info: ... okay on input line 32. LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 32. LaTeX Font Info: ... okay on input line 32. LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 32. LaTeX Font Info: ... okay on input line 32. LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 32. LaTeX Font Info: ... okay on input line 32. (/usr/share/texmf-dist/tex/context/base/mkii/supp-pdf.mkii [Loading MPS to PDF converter (version 2006.09.02).] \scratchcounter=\count317 \scratchdimen=\dimen281 \scratchbox=\box67 \nofMPsegments=\count318 \nofMParguments=\count319 \everyMPshowfont=\toks42 \MPscratchCnt=\count320 \MPscratchDim=\dimen282 \MPnumerator=\count321 \makeMPintoPDFobject=\count322 \everyMPtoPDFconversion=\toks43 ) (/usr/share/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty Package: epstopdf-base 2020-01-24 v2.11 Base part for package epstopdf Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4 85. (/usr/share/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) *geometry* driver: auto-detecting *geometry* detected driver: pdftex *geometry* verbose mode - [ preamble ] result: * driver: pdftex * paper: a4paper * layout: * layoutoffset:(h,v)=(0.0pt,0.0pt) * modes: * h-part:(L,W,R)=(113.81102pt, 369.88583pt, 113.81102pt) * v-part:(T,H,B)=(56.9055pt, 702.78308pt, 85.35826pt) * \paperwidth=597.50787pt * \paperheight=845.04684pt * \textwidth=369.88583pt * \textheight=702.78308pt * \oddsidemargin=41.54103pt * \evensidemargin=41.54103pt * \topmargin=-52.36449pt * \headheight=12.0pt * \headsep=25.0pt * \topskip=10.0pt * \footskip=30.0pt * \marginparwidth=65.0pt * \marginparsep=11.0pt * \columnsep=10.0pt * \skip\footins=9.0pt plus 4.0pt minus 2.0pt * \hoffset=0.0pt * \voffset=0.0pt * \mag=1000 * \@twocolumnfalse * \@twosidefalse * \@mparswitchfalse * \@reversemarginfalse * (1in=72.27pt=25.4mm, 1cm=28.453pt) LaTeX Font Info: Trying to load font information for U+msa on input line 34. (/usr/share/texmf-dist/tex/latex/amsfonts/umsa.fd File: umsa.fd 2013/01/14 v3.01 AMS symbols A ) LaTeX Font Info: Trying to load font information for U+msb on input line 34. (/usr/share/texmf-dist/tex/latex/amsfonts/umsb.fd File: umsb.fd 2013/01/14 v3.01 AMS symbols B ) LaTeX Font Info: Trying to load font information for U+rsfs on input line 34 . (/usr/share/texmf-dist/tex/latex/jknapltx/ursfs.fd File: ursfs.fd 1998/03/24 rsfs font definition file (jk) ) LaTeX Font Info: Trying to load font information for U+bbm on input line 42. (/usr/share/texmf-dist/tex/latex/bbm-macros/ubbm.fd File: ubbm.fd 1999/03/15 V 1.2 Font definition for bbm font - TH ) [1 {/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] Overfull \hbox (7.81883pt too wide) in paragraph at lines 144--145 []$\OT1/cmr/m/n/10 = [][] = [][] [][\OML/cmm/m/it/10 V[] \OT1/cmr/m/n/10 = 1] \ OMS/cmsy/m/n/10 ^^@ []\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 W[] \OT1/cmr/m/n/10 = 1 ] = [][](\OML/cmm/m/it/10 p[] \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 e[] \OT1/cm r/m/n/10 = [][] \OML/cmm/m/it/10 p[]\OT1/cmr/m/n/10 (1 \OMS/cmsy/m/n/10 ^^@ [] [2] Overfull \hbox (11.64793pt too wide) in paragraph at lines 204--205 []\OT1/cmr/bx/n/10 Definition 5. []\OT1/cmr/m/n/10 A pro-cess sat-is-fy-ing tho se hy-pothe-ses is called a $[]$ [] [3] [4] Overfull \hbox (4.31699pt too wide) in paragraph at lines 417--418 []$\OT1/cmr/m/n/10 = \OML/cmm/m/it/10 e[]e[]^^U\OT1/cmr/m/n/10 (\OML/cmm/m/it/1 0 t[] \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 s[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/1 0 e[]e[]^^U\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t[] \OMS/cmsy/m/n/10 ^^@ \OML/cmm/ m/it/10 s[]\OT1/cmr/m/n/10 ) [] \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 (1 \OMS/cms y/m/n/10 ^^@ \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 )$ [] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] Overfull \hbox (2.56673pt too wide) in paragraph at lines 1485--1486 \OT1/cmr/m/n/10 Choose $\OML/cmm/m/it/10 a > \OT1/cmr/m/n/10 0$ such that $\OML /cmm/m/it/10 ^^V\OT1/cmr/m/n/10 ((\OMS/cmsy/m/n/10 ^^@\OML/cmm/m/it/10 a; a\OT1 /cmr/m/n/10 ][]) \OML/cmm/m/it/10 < ^^O$ \OT1/cmr/m/n/10 [since $\OML/cmm/m/it/ 10 ^^V$ \OT1/cmr/m/n/10 is fi-nite] and for all $\OML/cmm/m/it/10 n$\OT1/cmr/m/ n/10 , $\OML/cmm/m/it/10 ^^V[]\OT1/cmr/m/n/10 (((\OMS/cmsy/m/n/10 ^^@\OML/cmm/m /it/10 a; a\OT1/cmr/m/n/10 ][])[]) \OML/cmm/m/it/10 < [] [18] [19] [20] [21] Overfull \hbox (6.7859pt too wide) in paragraph at lines 1764--1765 []\OT1/cmr/m/n/10 We con-struct a coun-terex-am-ple. Sup-pose $\OML/cmm/m/it/10 X; Y$ \OT1/cmr/m/n/10 have mean $0$, in-de-p-den-dent [so $\OML/cmm/m/it/10 E\ OT1/cmr/m/n/10 [\OML/cmm/m/it/10 XY\OT1/cmr/m/n/10 ] = [] [22] [23] [24] [25] Overfull \hbox (20.10469pt too wide) detected at line 2138 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ] = \OML/c mm/m/it/10 E [] \OT1/cmr/m/n/10 = [] \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cm m/m/it/10 X\U/bbm/m/n/10 1[]\OT1/cmr/m/n/10 ] = [] \OML/cmm/m/it/10 E\OT1/cmr/m /n/10 [\OML/cmm/m/it/10 X\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 B\OT1/cmr/m/n/10 ]\ OML/cmm/m/it/10 P\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 B\OT1/cmr/m/n/10 ) = [][] \O ML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OMS/cmsy/m/n/10 j\U/rsfs/m/ n/10 G\OT1/cmr/m/n/10 ] d\OML/cmm/m/it/10 P \OT1/cmr/m/n/10 = \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OMS/cms y/m/n/10 j\U/rsfs/m/n/10 G\OT1/cmr/m/n/10 ]] [] [26] [27] Overfull \hbox (6.50827pt too wide) in paragraph at lines 2228--2229 []\OT1/cmr/bx/n/10 Theorem 26 \OT1/cmr/m/n/10 (34.3)\OT1/cmr/bx/n/10 . []\OT1/c mr/m/n/10 If $\OML/cmm/m/it/10 X$ \OT1/cmr/m/n/10 is $\U/rsfs/m/n/10 G$\OT1/cmr /m/n/10 -measurable and $\OML/cmm/m/it/10 Y; XY \OMS/cmsy/m/n/10 2 \OML/cmm/m/i t/10 L[]\OT1/cmr/m/n/10 ( \OML/cmm/m/it/10 ; \U/rsfs/m/n/10 F\OML/cmm/m/it/10 ; \U/rsfs/m/n/10 P\OT1/cmr/ m/n/10 )$ then $\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 XY\OMS/cmsy /m/n/10 j\U/rsfs/m/n/10 G\OT1/cmr/m/n/10 ] = [] [28] [29] [30] [31] [32] Overfull \hbox (9.23212pt too wide) in paragraph at lines 2653--2653 []\OT1/cmr/bx/n/14.4 Section 36: Kol-mogorov's Ex-is-tence/Consistency The- [] [33] Overfull \hbox (7.94763pt too wide) in paragraph at lines 2804--2805 []\OT1/cmr/bx/n/10 Theorem 32. []\OT1/cmr/m/n/10 For a stochas-tic pro-cess wit h in-de-pen-dent sta-tion-ary in-cre-ment, $(\OMS/cmsy/m/n/10 ^^C\OT1/cmr/m/n/1 0 ) \OMS/cmsy/m/n/10 ([]) [] [34] Overfull \hbox (10.38895pt too wide) in paragraph at lines 2861--2862 []\OT1/cmr/m/n/10 Let $\OML/cmm/m/it/10 M$ \OT1/cmr/m/n/10 b the lin-ear trans- for-ma-tion that takes $\OMS/cmsy/m/n/10 h\OML/cmm/m/it/10 y[]; y[] \OMS/cmsy/m /n/10 ^^@ \OML/cmm/m/it/10 y[]; [] ; y[] \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 y[]\OMS/cmsy/m/n/10 i$ \OT1/cmr/m/n/10 to $\OMS/cmsy/m/n/10 h\OML/cmm/m/it/10 y[]; y[]; [] ; y[]\OMS/cmsy/m/n/10 i$ [] [35] [36] Overfull \hbox (1.9071pt too wide) in paragraph at lines 3053--3054 []\OT1/cmr/m/n/10 We com-pare $\OML/cmm/m/it/10 W []$ \OT1/cmr/m/n/10 to $\OML/ cmm/m/it/10 W\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ for $\OML/cm m/m/it/10 j \OT1/cmr/m/n/10 = 0\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1\OML/cmm/m/i t/10 ; \OT1/cmr/m/n/10 2\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 3$ to get $\OMS/cmsy /m/n/10 j\OML/cmm/m/it/10 W\OT1/cmr/m/n/10 ([]) \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m /it/10 W\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 j ^^T \OML/cmm/m/it/10 C [] \OMS/cmsy/m/n/10 ^^T [] [37] [38] [39] [40] (./Probability 564.aux) *********** LaTeX2e <2024-11-01> patch level 2 L3 programming layer <2025-01-18> *********** ) Here is how much of TeX's memory you used: 15403 strings out of 475171 314301 string characters out of 5767095 695915 words of memory out of 5000000 38123 multiletter control sequences out of 15000+600000 569666 words of font info for 83 fonts, out of 8000000 for 9000 14 hyphenation exceptions out of 8191 116i,15n,121p,759b,984s stack positions out of 10000i,1000n,20000p,200000b,200000s Output written on "Probability 564.pdf" (40 pages, 422419 bytes). PDF statistics: 258 PDF objects out of 1000 (max. 8388607) 167 compressed objects within 2 object streams 0 named destinations out of 1000 (max. 500000) 13 words of extra memory for PDF output out of 10000 (max. 10000000)