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\lhead{CS 555, FALL 2023}               
\rhead{Name: Hemanta K. Maji} %%% <-- REPLACE Hemanta K. Maji WITH YOUR NAME HERE

\begin{document}

\title{Homework 1}

\date{}

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\begin{enumerate}

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%%%%%%%%%%%% PROBLEM 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries Estimating $(1-x)$ using $\exp(\cdot)$ function.} 
For $x\in[0,1),$ we know that 
    $$\ln(1-x) = -x - \frac{x^2}2 - \frac{x^3}3 - \dotsi.$$
        
    \begin{enumerate} 
    \item {\bfseries (5 points)} 
    Prove that $1-x \leq \exp\left( - x - \frac{x^2}2 \right) .$
    
    {\bfseries Solution.} 
    
    \   %%% <-- ERASE THIS LINE AND WRITE YOUR SOLUTION HERE
    \newpage
    
    \item {\bfseries (5 points)} 
    For $x\in[0,1/2]$, prove that 
        $$1- x \geq \exp\left(-x-x^2\right) .$$
        
    {\bfseries Solution.} 
    
    \   %%% <-- ERASE THIS LINE AND WRITE YOUR SOLUTION HERE
    \end{enumerate}
\newpage 




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%%%%%%%%%%%% PROBLEM 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries Tight Estimations}
Provide meaningful upper and lower bounds for the following expressions.

    \begin{enumerate}
    \item {\bfseries (5 points)} 
    $S_n = \sum_{i = 1}^{\infty} i^{-\frac{13}{11}}.$

    Hint: Your upper and lower bounds should be constants.
    
    {\bfseries Solution.} 
    
    \   %%% <-- ERASE THIS LINE AND WRITE YOUR SOLUTION HERE  
    \newpage
    
    \item {\bfseries (10 points)} 
    $A_n = n!.$

    Hint: You may want to start by upper and lower bounding $S_n = \sum_{i=1}^n \ln i.$
    
    {\bfseries Solution.} 
    
    \   %%% <-- ERASE THIS LINE AND WRITE YOUR SOLUTION HERE  
    \newpage
    
    \item {\bfseries (10 points)} 
    $B_n = \binom{2n}{n}.$ 

    Hint: Note that $\binom{2n}{n} = \frac{(2n)!}{(n!)^2}.$
    
    {\bfseries Solution.} 
    
    \   %%% <-- ERASE THIS LINE AND WRITE YOUR SOLUTION HERE
    \end{enumerate} 
\newpage





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%%%%%%%%%%%% PROBLEM 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries Understanding Joint Distribution.}
Ten balls are to be tossed into five bins numbered $\{1, 2, 3, 4, 5\}$. 
Each ball is thrown into a bin uniformly and independently into the bins. 
For $i\in\{1,2,3,4,5\}$, let $X_i$ represent the number of balls that fall into bin $i$.

\begin{enumerate}
    \item {\bfseries (5 points)} 
    Find the (marginal) distribution of $X_1$ and compute its expected value. 

    {\bfseries Solution.} 
    
    \   %%% <-- ERASE THIS LINE AND WRITE YOUR SOLUTION HERE  

    \vspace{0.3\textheight} %%% <-- ERASE THIS LINE

    \item {\bfseries (3 points)}
    Find the expected value of $X_1+X_2+X_3$. 

    {\bfseries Solution.} 
    
    \   %%% <-- ERASE THIS LINE AND WRITE YOUR SOLUTION HERE

    \vspace{0.3\textheight} %%% <-- ERASE THIS LINE

    \newpage
    
    \item {\bfseries (7 points)}
    Find $\prob{[X_1 = 4 \vert X_1+X_2+X_3 = 7]}.$

    {\bfseries Solution.} 
    
    \   %%% <-- ERASE THIS LINE AND WRITE YOUR SOLUTION HERE  
\end{enumerate}
\newpage
    
    
    
    
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%%%%%%%%%%%% PROBLEM 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries Sending one bit.}

Alice intends to send a bit $b\in\{0,1\}$ to Bob. 
When Alice sends the bit, it goes through a series of $n$ relays before reaching Bob.
Each relay flips the received bit independently with probability $p$ before forwarding that bit to the next relay.

    \begin{enumerate}
    \item {\bfseries (5 points)} 
    % What is the probability that Bob receives the correct bit?
    Show that Bob will receive the correct bit with probability 
        $$\sum_{k = 0}^{\lfloor n/2 \rfloor} {\binom{n}{2k}} p^{2k} \cdot (1 - p)^{n - 2k}.$$
    Hint: Be careful that Alice could be sending either $0$ or $1$.
 
    {\bfseries Solution.}
    
    \   %%% <-- ERASE THIS LINE AND WRITE YOUR SOLUTION HERE

    
    \newpage
      
    \item {\bfseries (5 points)} 
    Let us consider an alternative way to calculate this probability.
    We say that the relay has {\em{bias}} $q$ if the probability it flips the bit is $(1-q)/2$.
    The bias $q$ is a real number between $-1$ and $+1$.
    Show that sending a bit through two relays with bias $q_1$ and $q_2$ is equivalent to sending a bit through a single relay with bias $q_1 \cdot q_2$.
    
    {\bfseries Solution.}
    
    \   %%% <-- ERASE THIS LINE AND WRITE YOUR SOLUTION HERE
    
    \vspace{0.35\textheight} %%% <-- ERASE THIS LINE
      
    \item {\bfseries (5 points)} 
    Prove that the probability you receive the correct bit when it passes through $n$ relays is $$\frac{1+(1-2p)^n}{2}.$$
   
    {\bfseries Solution.}
    
    \   %%% <-- ERASE THIS LINE AND WRITE YOUR SOLUTION HERE
    \end{enumerate}
\newpage




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%%%%%%%%%%%% PROBLEM 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries An Useful Estimate.}

For an integers $n$ and $t$ satisfying $0\leq t\leq n/2,$ define 
  $$ P_n(t) = \left(1-\frac1n\right)\left(1-\frac2n\right)\dotsi\left(1-\frac{t}n\right) $$
We will estimate the above expression.
({\footnotesize {\em Remark}: You shall see the usefulness of this estimation in the topic ``Birthday Bound'' that we shall cover in the forthcoming lectures.})

    \begin{enumerate}
    \item {\bfseries (13 points)}
    Show that 
        $$ \exp\left(-\frac{t^2}{2n} - \frac{t}{2n} - \frac{\Theta\left(t^3\right)}{6n^2}\right) 
        \geq P_n(t) \geq 
        \exp\left(-\frac{t^2}{2n} - \frac{t}{2n} -\frac{\Theta\left(t^3\right)}{3n^2}\right).$$
        
    {\bfseries Solution.}
    
    \   %%% <-- ERASE THIS LINE AND WRITE YOUR SOLUTION HERE
    
    \newpage
    
    \item {\bfseries (2 points)}
    When $t=\sqrt{2cn},$ where $c$ is a positive constant, the expression above is
      $$ P_n(t) = \exp\left( \; -c - \; \Theta\left(1/\sqrt n\right) \; \right).$$
    
    {\bfseries Solution.}
    
    \   %%% <-- ERASE THIS LINE AND WRITE YOUR SOLUTION HERE
    
    \end{enumerate}
\newpage


\end{enumerate}
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