\documentclass[10pt]{article}
\usepackage[margin=0.7in,top=0.5in]{geometry}
\usepackage[small]{titlesec}
\usepackage{palatino, mathpazo}
\usepackage{inconsolata}
\usepackage{amsmath, amssymb}
\usepackage{enumitem}
\usepackage{ulem}
\normalem
\newcommand{\ds}{\displaystyle}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\Var}{\operatorname{Var}}
\renewcommand{\P}{\mathbb{P}}
\setlength{\parindent}{0pt}
\setlength{\parskip}{1.5ex}
\pagenumbering{gobble}
\title{\sc Math 60 -- Homework 5}
\author{due Wednesday, May 2}
\date{}
\begin{document}
\maketitle
\textbf{Instructions:} This assignment is due at the \emph{beginning} of class. Staple your work together (do not just fold over the corner). Please write the questions in the correct order. If I cannot read your handwriting, you won't receive full credit. Every step of your answers must be fully justified to receive credit.
%\emph{You may use Wolfram Alpha or another similar tool to compute any necessary sums or integrals, and for your matrix calculations. If you have trouble with this, let me know.}\\
%\textbf{If you're using facts about distributions to answer the questions, be very clear about which distribution you're using to model that problem and why that distribution is appropriate.}
\underline{\textbf{It is considered cheating and a violation of the Honor Code to look for answers to these problems on the internet.}}
\begin{enumerate}
\item Assume $S$ and $T$ are independent discrete random variables both on the sample space $\N$ and $Q = S + T$. Find (and prove) an expression for the probability distribution function of Q (i.e., $\P(Q=k)$) for an arbitrary nonnegative integer $k$ in terms of the probability distribution functions for $S$ and $T$.
\item Let $X$ and $Y$ be independent binomial random variables, such that both $X$ and $Y$ have the same probability of success $p$ and have number of trials $N_X$ and $N_Y$.
\begin{enumerate}
\item Describe the random variable $Z = X+Y$, giving its probability distribution function, expected value, variance, and standard deviation.
\item Repeat part (a) for $W = X-Y$.
\end{enumerate}
\item Let $P_1,P_2,\ldots$ be iid Poisson random variables with rate $\lambda$.
\begin{enumerate}
\item Prove that $P_1+P_2$ is itself a Poisson random variable with rate $2\lambda$.
\item Prove that $P_1 + P_2 + \cdots + P_n$ is itself a Poisson random variable with rate $n\lambda$. You may use the fact that
\[
\ds\sum_{k=0}^n A^{n-k}{n \choose k} = (A+1)^n.
\]
\item Give an intuitive explanation for why (a) and (b) make sense.
\end{enumerate}
\item An airline finds that 4\% of the passengers that make reservations on a particular flight will not show up. Consequently, their policy is to sell $100$ reserved seats on a plane that has only 98 seats. Find the probability that every person who shows up will get a seat, using
\begin{enumerate}
\item a binomial distribution, then Wolfram Alpha or some other tool to get a numerical answer
\item a Poisson approximation, then a calculator or Wolfram Alpha to get a numerical answer
\end{enumerate}
\item A baker blends 600 raisins and 400 chocolate chips into a dough mix and, from this, makes 500 cookies. Answer the following questions with both binomial and Poisson distributions, as in the previous question.
\begin{enumerate}
\item Find the probability that a randomly picked cookie will have no raisins.
\item Find the probability that a randomly picked cookie will have exactly two chocolate chips.
\item Find the probability that a randomly picked cookie will have at least two bits (raisins or chips) in it.
\end{enumerate}
\item One way to get out of jail in Monopoly is to roll doubles (two-of-a-kind) on a pair of fair six-sided dice. You get one attempt to roll doubles on each turn. Let $X$ be the number of turns required to get out of jail. Ignoring the Monopoly rule that says you have to pay to leave jail if you fail three times, find $\P(X=k)$, $\E[X]$, and $\Var(X)$.
\end{enumerate}
\end{document}