\documentclass[11pt]{article}
\input{headers2}

\usepackage{fancyhdr}   
\pagestyle{fancy}      
\lhead{}               
\rhead{Name: Hemanta K. Maji} %%% <-- YOUR NAME HERE

\usepackage[strict]{changepage}  
\newcommand{\nextoddpage}{\checkoddpage\ifoddpage{\ \newpage\ \newpage}\else{\ \newpage}\fi}  


\begin{document}

\title{Homework 2\\(150 points)}

\date{}

\maketitle 

\thispagestyle{fancy}  
\pagestyle{fancy}  

    




\begin{enumerate}

%%% Problem 1 
\item {\bfseries (10 + 10 + 20 points) An Interesting Concentration.}
  Let $\X$ be the random variable over the sample space $\{1, 2, \dotsc\}$ such that
  $\probX{\X=i}=2^{-i}$. 
  \begin{enumerate}
  \item Compute $\mu = \EX{\X}$. 
  \item Define $\Y= \X-\mu$. 
    For $0\leq h\leq \ln 2$, compute $\EX{\exp(h\Y)}$. 
  \item Define $\S_n = \Y\p1+\dotsi+\Y\p n$. 
    Find the concentration bound for $\probX{\S_n\geq t}$ using the technique of Chernoff bound. 
  \end{enumerate}  
 
\nextoddpage
%%% Problem 2
\item {\bfseries (10 + 10 + 20 points) Concentration of Sum of Poisson Distribution.} 
  Let $\X$ be the random variable over the sample space $\{0,1,\dotsc\}$ such that
  $\probX{\X=i}= \exp(-\mu)\frac{\mu^i}{i!}$.
  \begin{enumerate}
  \item Prove that $\EX{\X}=\mu$. 
  \item Define $\Y=\X-\mu$. 
    For positive $h$, compute $\EX{\exp(h\Y)}$. 
  \item Define $\S_n = \Y\p1 + \dotsi + \Y\p n$. 
    Find the concentration bound for $\probX{\S_n\geq t}$ using the technique of Chernoff bound. 
    (You might find it useful to use a variable $m$ such that $m=n\mu$ in the final bound.)
  \end{enumerate}  
  
  
\nextoddpage
%%% Problem 3
\item {\bfseries (10 + 10 points) Coin Tossing.}
  Let $\X$ be the uniform distribution over the sample space $\{0,1\}$. 
  \begin{enumerate}
  \item Let $S_n = \X\p1+\dotsi+\X\p n$. 
    Given a fixed values of $m$, how will you choose $n$ such that $\probX{\S_n\geq m} \leq (1-\eps)$? 
  \item Use the above result to prove the concentration bound in Problem 1 part c. 
  \end{enumerate} 
  
\nextoddpage
%%% Problem 4
\item {\bfseries (10 points) Concentration of Matrix rank.} 
  Let $\M$ be a distribution over $n\times n$ matrices, where each element is selected uniformly and independently at random from the set $\Omega$. 
  State and prove a concentration bound for the rank of $\M$ around its median or mean. 
  
  
  
\nextoddpage
%%% Problem 5
\item {\bfseries (40 points) Prefix-sum of Coins are Close to their respective Mean.} 
  Let $\X$ be a distribution over \zo such that $\probX{\X=1}=p$ and $\probX{\X=0}=(1-p)$. 
  We consider the sum $S_n=\X\p1 + \dotsi+\X\p n$. \\
  
  \hrule 
  
  Chernoff-Hoeffding's bound states the following. 
  It says that the probability of the sum $S_n$ exceeding the expectation by $t$ is very small.
  For example, we can say that
  $$\probX{\S_n\geq pn + t}\leq \exp\left(-{2t^2}/n\right)$$
  Intuitively, suppose we reject any outcome of the coins such that $\S_n\geq pn+t$. 
  Then, this bound says that the probability of rejecting is at most $\exp\left(-{2t^2}/n\right)$. \\
  
  \hrule
  
  We want to claim that ``$\S_n$ never exceeded the expectation in any prefix.'' 
  Let me elaborate. 
  Suppose we reject any coin such that $\S_i \geq p\cdot i+t$ for any $i\in\{1,\dotsc,n\}$. 
  Formally, we reject if there exists $i\in\{1,\dotsc,n\}$ such that $\S_i\geq p\cdot i+t$. 
  Note that this rejection rule is {\em more stringent} than the previous rejection criterion. 
  Our goal is to prove that this rejection probability is small. 
  In particular, prove that
  $$\probX{\exists i\in\{1,\dotsc,n\} ~\st~ \S_i\geq p\cdot i+t}\leq \exp\left(-{2t^2}/{n}\right)$$
  Isn't this amazing? 
  This bound is identical to the Chernoff-Hoeffding bound! 
  
  

\nextoddpage   
%%% Extra Credit
\item {\bfseries (Extra Credit) New bounds for Hoeffding's Lemma.} 
  Surprise me with a new statement/proof of Hoeffding's Lemma! 


\end{enumerate}









\end{document}
