From cd062e333d3b242b4d3fb0a0835f040ebe924dc0 Mon Sep 17 00:00:00 2001 From: Uzair Mohammed Date: Wed, 7 Feb 2024 00:56:19 -0800 Subject: [PATCH] HW0 - Did some work, and added unanswered questions. Homework 0 is incomplete --- main.aux | 3 +- main.fdb_latexmk | 32 ++++++----- main.fls | 47 ++++++++-------- main.log | 137 ++++++++++++++++++++++++++++------------------- main.pdf | Bin 89941 -> 149414 bytes main.synctex.gz | Bin 8572 -> 17038 bytes main.tex | 57 +++++++++++++++++--- titlePage.aux | 1 + 8 files changed, 177 insertions(+), 100 deletions(-) diff --git a/main.aux b/main.aux index ff7f954..81b430d 100644 --- a/main.aux +++ b/main.aux @@ -3,4 +3,5 @@ \@writefile{toc}{\contentsline {section}{\numberline {0}Homework 0}{2}{}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\numberline {0.1}About You}{2}{}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\numberline {0.2}Optimization}{2}{}\protected@file@percent } -\gdef \@abspage@last{3} +\@writefile{toc}{\contentsline {subsection}{\numberline {0.3}Probability}{3}{}\protected@file@percent } +\gdef \@abspage@last{5} diff --git a/main.fdb_latexmk b/main.fdb_latexmk index 62ba88d..a883df7 100644 --- a/main.fdb_latexmk +++ b/main.fdb_latexmk @@ -1,35 +1,39 @@ # Fdb version 4 -["pdflatex"] 1707289023.61851 "c:/Users/uzair/OneDrive/Documents/CSC 665/main.tex" "main.pdf" "main" 1707289024.35903 0 +["pdflatex"] 1707295945.19537 "c:/Users/uzair/OneDrive/Documents/CSC 665/main.tex" "main.pdf" "main" 1707295945.95425 0 "C:/Users/uzair/AppData/Local/MiKTeX/fonts/map/pdftex/pdftex.map" 1707288871 80909 eab91d9745dd2edfd62a31d53cd5fe15 "" "C:/Users/uzair/AppData/Local/MiKTeX/fonts/pk/ljfour/jknappen/ec/dpi600/tcrm1000.pk" 1707288887 11548 1854d2b3451e3d517cccd4e0b4daead1 "" "C:/Users/uzair/AppData/Local/MiKTeX/miktex/data/le/pdftex/pdflatex.fmt" 1707288883 24210786 eea8c83a2618bf979528f763e32e98c9 "" "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/tfm/jknappen/ec/tcrm1000.tfm" 993058908 1436 c7f957a372ef2fbe93c0982f96625e12 "" + "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1233951848 1004 54797486969f23fa377b128694d548df "" + 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"C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1247593067 30251 6afa5cb1d0204815a708a080681d4674 "" "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmmi10.pfb" 1247593067 36299 5f9df58c2139e7edcf37c8fca4bd384d "" + "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmmi7.pfb" 1247593067 36281 c355509802a035cadc5f15869451dcee "" "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr10.pfb" 1247593067 35752 024fb6c41858982481f6968b5fc26508 "" "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr12.pfb" 1247593067 32722 d7379af29a190c3f453aba36302ff5a9 "" + "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr17.pfb" 1247593066 32362 bc3f3eec7ab7d65fe700963d4017d32c "" "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr7.pfb" 1247593067 32762 7fee39e011c23b3589931effd97b9702 "" + "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1247593067 32569 5e5ddc8df908dea60932f3c484a54c0d "" + "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy7.pfb" 1247593066 32716 08e384dc442464e7285e891af9f45947 "" "C:/Users/uzair/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmti10.pfb" 1247593067 37944 359e864bd06cde3b1cf57bb20757fb06 "" - "C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/context/base/mkii/supp-pdf.mkii" 1580390158 71627 94eb9990bed73c364d7f53f960cc8c5b "" - "C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/00miktex/epstopdf-sys.cfg" 1616063685 584 2a1075dd71571459f59146da9f7502ad "" + "C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/amsmath/amsbsy.sty" 1686928188 2222 499d61426192c39efd8f410ee1a52b9c "" + "C:/Users/uzair/AppData/Local/Programs/MiKTeX/tex/latex/amsmath/amsgen.sty" 1686928187 4173 82ac04dfb1256038fad068287fbb4fe6 "" + 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1707289022 2594 54fa70c93370d22be6f274197a03a703 "" - "titlePage.aux" 1707289024 425 3c34eeb34cfa1d08b86639e0e4c6ded9 "pdflatex" + "c:/Users/uzair/OneDrive/Documents/CSC 665/main.tex" 1707295944 5097 898378ef481e2be019a7f5f95119972e "" + "main.aux" 1707295945 464 6dfe8e55ee4a56697c8e62c22d665a53 "pdflatex" + "main.tex" 1707295944 5097 898378ef481e2be019a7f5f95119972e "" + "titlePage.aux" 1707295945 456 c979711f53deacf2b09031977e556db8 "pdflatex" "titlePage.tex" 1707288498 394 ccdbd50244b3194dbad72dd1c7995bf0 "" (generated) "main.aux" diff --git a/main.fls b/main.fls index bd8b250..dab785a 100644 --- a/main.fls +++ b/main.fls @@ -7,20 +7,16 @@ INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\base\article.cls INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\base\size10.clo INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\base\size10.clo INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\base\size10.clo -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics\graphicx.sty -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics\graphicx.sty -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics\keyval.sty -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics\keyval.sty -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics\graphics.sty -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics\graphics.sty -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics\trig.sty -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics\trig.sty -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics-cfg\graphics.cfg -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics-cfg\graphics.cfg -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics-cfg\graphics.cfg -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics-def\pdftex.def -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics-def\pdftex.def -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\graphics-def\pdftex.def +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\amsmath\amsmath.sty +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\amsmath\amsmath.sty +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\amsmath\amsopn.sty +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\amsmath\amstext.sty +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\amsmath\amstext.sty +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\amsmath\amsgen.sty +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\amsmath\amsgen.sty +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\amsmath\amsbsy.sty +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\amsmath\amsbsy.sty +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\amsmath\amsopn.sty INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\l3backend\l3backend-pdftex.def INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\l3backend\l3backend-pdftex.def INPUT .\main.aux @@ -30,14 +26,6 @@ INPUT .\titlePage.aux INPUT .\titlePage.aux INPUT titlePage.aux OUTPUT main.aux -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\context\base\mkii\supp-pdf.mkii -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\context\base\mkii\supp-pdf.mkii -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\context\base\mkii\supp-pdf.mkii -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\epstopdf-pkg\epstopdf-base.sty -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\epstopdf-pkg\epstopdf-base.sty -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\00miktex\epstopdf-sys.cfg -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\00miktex\epstopdf-sys.cfg -INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex\latex\00miktex\epstopdf-sys.cfg INPUT .\titlePage.tex INPUT .\titlePage.tex OUTPUT titlePage.aux @@ -51,24 +39,39 @@ OUTPUT main.pdf INPUT C:\Users\uzair\AppData\Local\MiKTeX\fonts\map\pdftex\pdftex.map INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmr12.tfm INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmbx12.tfm +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\cmextra\cmex7.tfm +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\cmextra\cmex7.tfm INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmti10.tfm INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmbx12.tfm INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\tfm\jknappen\ec\tcrm1000.tfm +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmbx10.tfm INPUT main.aux INPUT .\titlePage.aux INPUT .\titlePage.aux INPUT titlePage.aux INPUT C:\Users\uzair\AppData\Local\MiKTeX\fonts\pk\ljfour\jknappen\ec\dpi600\tcrm1000.pk INPUT C:\Users\uzair\AppData\Local\MiKTeX\fonts\pk\ljfour\jknappen\ec\dpi600\tcrm1000.pk +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmbx10.pfb +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmbx10.pfb INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmbx12.pfb INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmbx12.pfb +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmex10.pfb +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmex10.pfb INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmmi10.pfb INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmmi10.pfb +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmmi7.pfb +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmmi7.pfb INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmr10.pfb INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmr10.pfb INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmr12.pfb INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmr12.pfb +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmr17.pfb +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmr17.pfb INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmr7.pfb INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmr7.pfb +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy10.pfb +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy10.pfb +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy7.pfb +INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy7.pfb INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmti10.pfb INPUT C:\Users\uzair\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmti10.pfb diff --git a/main.log b/main.log index 5be689a..503bcdd 100644 --- a/main.log +++ b/main.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.25 (MiKTeX 24.1) (preloaded format=pdflatex 2024.2.6) 6 FEB 2024 22:57 +This is pdfTeX, Version 3.141592653-2.6-1.40.25 (MiKTeX 24.1) (preloaded format=pdflatex 2024.2.6) 7 FEB 2024 00:52 entering extended mode restricted \write18 enabled. file:line:error style messages enabled. @@ -23,28 +23,74 @@ File: size10.clo 2023/05/17 v1.4n Standard LaTeX file (size option) \abovecaptionskip=\skip48 \belowcaptionskip=\skip49 \bibindent=\dimen140 -) (C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/graphics\graphicx.sty -Package: graphicx 2021/09/16 v1.2d Enhanced LaTeX Graphics (DPC,SPQR) - (C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/graphics\keyval.sty -Package: keyval 2022/05/29 v1.15 key=value parser (DPC) -\KV@toks@=\toks17 -) (C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/graphics\graphics.sty -Package: graphics 2022/03/10 v1.4e Standard LaTeX Graphics (DPC,SPQR) - (C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/graphics\trig.sty -Package: trig 2021/08/11 v1.11 sin cos tan (DPC) -) (C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/graphics-cfg\graphics.cfg -File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration +) (C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/amsmath\amsmath.sty +Package: amsmath 2023/05/13 v2.17o AMS math features +\@mathmargin=\skip50 + +For additional information on amsmath, use the `?' option. +(C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/amsmath\amstext.sty +Package: amstext 2021/08/26 v2.01 AMS text + (C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/amsmath\amsgen.sty +File: amsgen.sty 1999/11/30 v2.0 generic functions +\@emptytoks=\toks17 +\ex@=\dimen141 +)) (C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/amsmath\amsbsy.sty +Package: amsbsy 1999/11/29 v1.2d Bold Symbols +\pmbraise@=\dimen142 +) (C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/amsmath\amsopn.sty +Package: amsopn 2022/04/08 v2.04 operator names ) -Package graphics Info: Driver file: pdftex.def on input line 107. - (C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/graphics-def\pdftex.def -File: pdftex.def 2022/09/22 v1.2b Graphics/color driver for pdftex -)) -\Gin@req@height=\dimen141 -\Gin@req@width=\dimen142 +\inf@bad=\count195 +LaTeX Info: Redefining \frac on input line 234. +\uproot@=\count196 +\leftroot@=\count197 +LaTeX Info: Redefining \overline on input line 399. +LaTeX Info: Redefining \colon on input line 410. +\classnum@=\count198 +\DOTSCASE@=\count199 +LaTeX Info: Redefining \ldots on input line 496. +LaTeX Info: Redefining \dots on input line 499. +LaTeX Info: Redefining \cdots on input line 620. +\Mathstrutbox@=\box51 +\strutbox@=\box52 +LaTeX Info: Redefining \big on input line 722. +LaTeX Info: Redefining \Big on input line 723. +LaTeX Info: Redefining \bigg on input line 724. +LaTeX Info: Redefining \Bigg on input line 725. +\big@size=\dimen143 +LaTeX Font Info: Redeclaring font encoding OML on input line 743. +LaTeX Font Info: Redeclaring font encoding OMS on input line 744. +\macc@depth=\count266 +LaTeX Info: Redefining \bmod on input line 905. +LaTeX Info: Redefining \pmod on input line 910. +LaTeX Info: Redefining \smash on input line 940. +LaTeX Info: Redefining \relbar on input line 970. +LaTeX Info: Redefining \Relbar on input line 971. +\c@MaxMatrixCols=\count267 +\dotsspace@=\muskip16 +\c@parentequation=\count268 +\dspbrk@lvl=\count269 +\tag@help=\toks18 +\row@=\count270 +\column@=\count271 +\maxfields@=\count272 +\andhelp@=\toks19 +\eqnshift@=\dimen144 +\alignsep@=\dimen145 +\tagshift@=\dimen146 +\tagwidth@=\dimen147 +\totwidth@=\dimen148 +\lineht@=\dimen149 +\@envbody=\toks20 +\multlinegap=\skip51 +\multlinetaggap=\skip52 +\mathdisplay@stack=\toks21 +LaTeX Info: Redefining \[ on input line 2953. +LaTeX Info: Redefining \] on input line 2954. ) (C:\Users\uzair\AppData\Local\Programs\MiKTeX\tex/latex/l3backend\l3backend-pdftex.def File: l3backend-pdftex.def 2024-01-04 L3 backend support: PDF output (pdfTeX) -\l__color_backend_stack_int=\count195 -\l__pdf_internal_box=\box51 +\l__color_backend_stack_int=\count273 +\l__pdf_internal_box=\box53 ) (main.aux (titlePage.aux)) \openout1 = `main.aux'. @@ -62,56 +108,37 @@ LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 6. 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z35k{_hj^l*AOMODU09iQJftO~5aiGw;a#r5Iy&}w5p-hFn|-pbo?5lIB0fg4K`#;G zKvTiX(Bli`eoBhZj5poXl@~LsjJOhdO~p7B!Vj~bQZf&QN-ESz>MQjFCBZI0M3%F~ zqx?{Y866D?@Jk-j4fh^qrQ?J8Dq@RSPj9OD+_I=nn|+1H#EsaXGMQKA{Xeh#P9@$b z&F}>@Z99hcSrvBR{|`fLcr^axXpmVz~?CP)x}k>7`B)J6nSxbvUhZ7 vzq+z!7k+WO0jv30cr(99q>n|{jhQPcKtryu=jXrg9^m-o@xP3F@ZkRd0T0NB diff --git a/main.tex b/main.tex index 1b57b5b..51a70da 100644 --- a/main.tex +++ b/main.tex @@ -1,5 +1,5 @@ \documentclass[titlepage]{article} -\usepackage{graphicx} % Required for inserting images +\usepackage{amsmath} \setcounter{section}{-1} @@ -26,7 +26,7 @@ When it's time to submit an assignment, I will export this document as a PDF fil \newpage \section{Homework 0} -\textit{By turning in this assignment, I agree to abide by SFSU’s academic integrity code and declare that all of my solutions are my own work: } +\textit{By turning in this assignment, I agree to abide by SFSU's academic integrity code and declare that all of my solutions are my own work: } \subsection{About You} \begin{enumerate} @@ -42,18 +42,59 @@ When it's time to submit an assignment, I will export this document as a PDF fil \item Data Structures \end{itemize} This list consists of the courses I think are relevant to this class; I've taken other CS courses, of course. - \item Yes, I am okay with being called on. I won't always know the answer, and I might embarrass myself sometimes, but that's alright because I believe it will force me to pay more attention and learn better, and as a result be more prepared for assignments and exams. + \item [c.] Yes, I am okay with being called on. I won't always know the answer, and I might embarrass myself sometimes, but that's alright because I believe it will force me to pay more attention and learn better, and as a result be more prepared for assignments and exams. \end{enumerate} \subsection{Optimization} \begin{itemize} \item [a.] Let a, b, and c be positive real numbers. Consider the quadratic function - $$ - f(\theta) = a{\theta}^2 + b\theta + c - $$ - Note that $\theta$ here is a real number. What value of $\theta$ minimizes $f(\theta)$? - Solution: + $$f(\theta) = a{\theta}^2 + b\theta + c$$ + Note that $\theta$ here is a real number. What value of $\theta$ minimizes $f(\theta)$?\\[0.1in] + Solution: + Since this is a positive parabola, we can use the formula for the vertex of a parabola, $\frac{-b}{2a}$, to find the minimum:\\ + $$ + \boxed{\theta = \frac{-b}{2a}} + $$ + \item [b.] Let $x_1,\dots,x_n$ be real numbers. Consider the quadratic function + $$g(\theta) = \sum^n_{i=1}(\theta - x_i)^2.$$ + What value of $\theta$ minimizes $g(\theta)$? + + {\LARGE{TODO}} + + \item [c.] Let $x_1,\dots,x_n$ again be real numbers, and let $w_1,\dots,w_n$ be positive real numbers that we can interpret as representing the importance of each of the $x_i$'s. Consider the weighted quadratic function + $$h(\theta) = \sum_{i=1}^{n}w_i(\theta - x_i)^2.$$ + What value of $\theta$ minimizes $h(\theta)$? + + \item [d.] What issue could arise in the minimization of $h$ if some of the $w_i$'s are negative? +\end{itemize} + +\subsection{Probability} + +\begin{itemize} + \item [a.] Consider a standard 52-card deck of cards with 13 card values (Ace, King, Queen, Jack, and + 2-10) in each of the four suits (clubs, diamonds, hearts, spades). If a card is drawn at random, what + is the probability that it is a spade or a two? + + \item [b.] Two factories — Factory A and Factory B — design batteries to be used in mobile phones. + Factory A produces 60\% of all batteries, and Factory B produces the other 40\%. 2\% of Factory A's + batteries have defects, and 4\% of Factory B's batteries have defects. What is the probability that a + battery is both made by Factory A and defective? + + \item [c.] Consider the following (made up) facts about COVID incidence and testing: + \begin{itemize} + \item In the absence of any special information, the probability that a person has COVID is 1\%. + \item If a person has COVID, the probability that a test will correctly read positive is 80\%. + \item If a person does not have COVID, the probability that a test will incorrectly produce a false positive is 10\%. + \end{itemize} + + Suppose you take a COVID test and it reads positive. Given the facts above, what is the probability +that you have COVID? + + \item [d. ] Suppose you repeatedly roll a fair six-sided die until you roll a 1 (and then you stop). Every + time you roll a 3, you win a points, and every time you roll a 6, you lose b points. You do not win or + lose any points if you roll a 2, 4, or 5. What is the expected number of points (as a function of a and + b) you will have when you stop? \end{itemize} \end{document} \ No newline at end of file diff --git a/titlePage.aux b/titlePage.aux index 2bf9266..46c537d 100644 --- a/titlePage.aux +++ b/titlePage.aux @@ -16,4 +16,5 @@ \setcounter{subparagraph}{0} \setcounter{figure}{0} \setcounter{table}{0} +\setcounter{parentequation}{0} }