%------------------------------------------------------------------------------ %\documentclass[reqno]{amsart} \documentclass[12pt]{amsart} %\setcounter{page}{6} \usepackage[top=1in, bottom=1in, left=1in, right=1in]{geometry} \usepackage[colorlinks=true, urlcolor=blue]{hyperref} \usepackage{amssymb} \usepackage{amsmath,mathrsfs} %\oddsidemargin -0in \evensidemargin .5in %\topmargin=1.25in %\headheight 10pt \headsep 10pt \footheight 10pt \footskip 24pt %\textheight 10in \textwidth 6.5in \columnsep 10pt \columnseprule 0pt %\font\namefont=cmr10 scaled\magstep2 \font\namefont=cmr8 scaled\magstep2 \newcommand{\myindent}{\leftskip=.4in} %\voffset=-.75in \parskip=11pt %extra vertical distance for new paragraph \parindent=0in %\newtheorem{theorem}{Theorem}[section] %\newtheorem{lemma}[theorem]{Lemma} %\newtheorem{lemma}{Lemma}[section] %\newtheorem{theorem}{Theorem}[section] %\newtheorem{lemma}[theorem]{Lemma} %\newtheorem{prop}[theorem]{Proposition} %\newtheorem{cor}[theorem]{Corollary} %\newtheorem{conj}{Conjecture} \usepackage{graphicx} \usepackage{color} \usepackage{subfigure} \usepackage{amssymb} \usepackage{amsmath} \usepackage{colonequals} \usepackage{hyperref} %\usepackage{showlabels} %\usepackage[all]{xypic} %\entrymodifiers={+!!<0pt,\fontdimen22\textfont2>} \theoremstyle{definition} \newtheorem{exercise}{Exercise} \newtheorem{remark}{Remark} \renewcommand{\setminus}{\smallsetminus} \addtolength{\footskip}{17pt} %\numberwithin{table}{section} \renewcommand{\subset}{\subseteq} \renewcommand{\supset}{\supseteq} \renewcommand{\epsilon}{\varepsilon} \newcommand{\abs}[1]{\left|#1\right|} % Absolute value notation \newcommand{\absf}[1]{|#1|} % small absolute value signs \newcommand{\vnorm}[1]{\left|\left|#1\right|\right|} % norm notation \newcommand{\vnormf}[1]{||#1||} % norm notation, forced to be small \newcommand{\im}[1]{\mbox{im}#1} % Pieces of English for math mode \newcommand{\tr}[1]{\mbox{tr}#1} \newcommand{\Proj}[1]{\mbox{Proj}#1} \newcommand{\Vol}[1]{\mbox{Vol}#1} \newcommand{\Z}{\mathbf{Z}} % Blackboard notation \newcommand{\N}{\mathbf{N}} \newcommand{\E}{\mathbf{E}} \renewcommand{\P}{\mathbf{P}} \newcommand{\R}{\mathbf{R}} \newcommand{\C}{\mathbf{C}} \newcommand{\Q}{\mathbf{Q}} \newcommand{\figoneawidth}{.5\textwidth} % Image formatting parameters \newcommand{\lbreak}{\\} % Linebreak \newcommand{\italicize}[1]{\textit {#1}} % formatting commands for bibliography %\newcommand{\embolden}[1]{\textbf {#1}} \newcommand{\embolden}[1]{{#1}} \newcommand{\undline}[1]{\underline {#1}} \newcommand{\e}{\varepsilon} \renewcommand{\epsilon}{\varepsilon} \newcommand{\Samplespace}{\mathcal{C}} %consider changing this in next revision \newcommand{\samplept}{c} %\renewcommand{\colonequals}{=} \thispagestyle{empty} \begin{document} \bigskip Graduate Applied Probability II\hfill Steven Heilman\\ \noindent\rule{6.5in}{0.4pt} %\vspace{.2cm} Please provide complete and well-written solutions to the following exercises. Due January 29, 12PM noon PST, to be uploaded as a single PDF document to blackboard (under the Assignments tab). \vspace{.5cm} \begin{center} {\Large Homework 1} \end{center} \vspace{.5cm} \begin{exercise} Take the Spring 2017 and August 2017 qual exams for Math 505A, available here: \href{https://dornsife.usc.edu/assets/sites/990/docs/Applied_Probability/Applied_Probability_2012-2018.pdf}{https://dornsife.usc.edu/assets/sites/990/docs/Applied Probability/Applied Probability 2012-2018.pdf} (Record your answers, and submit them as part of this homework.) \end{exercise} \begin{exercise}[\embolden{Continuity of a Probability Law}]\label{exercise0123} Let $\P$ be a probability law on a sample space $\Samplespace$. Let $A_{1},A_{2},\ldots$ be sets in $\Samplespace$ which are increasing, so that $A_{1}\subset A_{2}\subset\cdots$. Then $$\lim_{n\to\infty}\P(A_{n})=\P(\cup_{n=1}^{\infty}A_{n}).$$ In particular, the limit on the left exists. Similarly, let $A_{1},A_{2},\ldots$ be sets in $\Samplespace$ which are decreasing, so that $A_{1}\supset A_{2}\supset\cdots$. Then $$\lim_{n\to\infty}\P(A_{n})=\P(\cap_{n=1}^{\infty}A_{n}).$$ \end{exercise} \begin{exercise}\label{exercise8.1z} Let $0