\documentclass[12pt]{amsart}
\usepackage{amssymb}
\usepackage{enumitem}
\usepackage[paperheight=11in,paperwidth=8.5in,left=1in,top=1in,right=1in,bottom=1in]{geometry}
\usepackage{hyperref}

\newcommand{\prob}{\mathbb{P}}
\newcommand{\E}{\mathbb{E}}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}



\begin{document}
\title{MATH 180A Homework \#3}
\author{Fall 2020}
\maketitle

Due date: \textbf{Friday 10/23/2020 11:59 PM} (via \href{https://www.gradescope.com/}{Gradescope})

\bigskip{}

\begin{itemize} \setlength{\itemsep}{10pt}
\item[1.] (\emph{ASV\footnotemark, Exercise 2.18}) - 2 points.
  \footnotetext{\emph{Introduction to Probability}, by David F. Anderson, Timo Sepp\"{a}l\"{a}inen, and Benedek Valk\'{o}}
  \\
  We choose a number from the set $\{10, 11, 12, \ldots, 99\}$ uniformly at random.
  \begin{itemize}
  \item[(a)] Let $X$ be the first digit and $Y$ the second digit of the chosen number. Show that $X$ and $Y$ are independent random variables.
  \item[(b)] Let $X$ be the first digit of the chosen number and $Z$ the sum of the two digits. Show that $X$ and $Z$ are not independent.
  \end{itemize}


\item[2.] (\emph{ASV, Exercise 1.36}) - 2 points.
  \begin{itemize}
  \item[(a)] Let $(X, Y)$ denote a uniformly chosen random point inside the unit square
    \[
      [0, 1]^2 = [0, 1] \times [0, 1] = \{(x, y): 0 \leq x, y \leq 1\}.
    \]
  Let $0 \leq a < b \leq 1$. Find the probability $\prob(a < X < b)$, that is, the probability that the $x$-coordinate $X$ of the chosen point lies in the interval $(a, b)$.  
  \item[(b)] What is the probability $\prob(|X - Y| \leq 1/4)$?
  \end{itemize}

\item[3.] (\emph{ASV, Exercise 3.20}) - 2 points.
  \\
  Let $c > 0$ and $X \sim \operatorname{Unif}[0,c]$. Show that the random variable $Y = c - X$ has the same cumulative distribution function as $X$ and hence also the same density function.
  
\item[4.] (\emph{ASV, Exercise 3.39}) - 2 points.
  \\
  Parts (a) and (b) ask for an example of a random variable $X$ whose cumulative distribution function $F(x)$ satisfies $F(1) = 1/3$, $F(2) = 3/4$, and $F(3) = 1$.
  \begin{itemize}
  \item[(a)] Make $X$ discrete and give its probability mass function.
  \item[(b)] Make $X$ continuous and give its probability density function.
  \end{itemize}
  
\item[5.] (\emph{ASV, Exercise 3.41}) - 3 points.
  \\
  We produce a random real number $X$ through the following two-stage experiment. First roll a fair die to get an outcome $Y$ in the set $\{1, 2, \ldots, 6\}$. Then, if $Y = k$, choose $X$ uniformly in the interval $(0, k]$. Find the cumulative distribution function $F(s)$ and the probability density function $f(s)$ of $X$ for $3 < s < 4$.

\item[6.] (\emph{ASV, Exercise 3.46}) - 3 points.
  \\
  A stick of length $\ell$ is broken at a uniformly chosen random location. We denote the length of the smaller piece by $X$.
  \begin{itemize}
  \item[(a)] Find the cumulative distribution function of $X$.
  \item[(b)] Find the probability density function of $X$.
  \end{itemize}

\item[7.] (\emph{ASV, Exercise 2.20}) - 3 points.
  \\
  A fair die is rolled repeatedly. Use precise notation of probabilities of events and random variables for the solutions to the questions below.
  \begin{itemize}
  \item[(a)] Write down a precise sum expression for the probability that the first five rolls give a three at most two times.
  \item[(b)] Calculate the probability that the first three does not appear before the fifth roll.
  \item[(c)] Calculate the probability that the first three appears before the twentieth roll but not before the fifth roll.
  \end{itemize}

\item[8.] (\emph{ASV, Exercise 2.21}) - 3 points.
  \\
  Jane must get at least three of the four problems on the exam correct to get an A. She has been able to do $80\%$ of the problems on old exams, so she assumes that the probability she gets any problem correct is $0.8$. She also assumes that the results on different problems are independent.
  \begin{itemize}
  \item[(a)] What is the probability she gets an A?
  \item[(b)] If she gets the first problem correct, what is the probability she gets an A?
  \end{itemize} 
\end{itemize}



\end{document}

%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% End:
