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\lhead{CS 592--ATK, SPRING 2022}               
\rhead{Name: Type your name here} %%% <-- REPLACE Hemanta K. Maji WITH YOUR NAME HERE

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\title{Homework 2}

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\begin{enumerate}

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%%%%%%%%%%%% PROBLEM 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries Sum of an Interesting Random Variable.} (20 points)
Let $\X$ be the random variable over the set of all natural numbers $\{ 1, 2, 3, \dotsc \}$ such that, for any natural number $i$, we have 
  $$\probX{\X=i}=2^{-i}.$$
  Let $\S_n = \X\p1+\X\p2+\dotsi+\X\p n$, where $\X\p1,\X\p2,\dotsc,\X\p n$ are independent and identical to $\X$. 
  \begin{itemize}
  \item (5 points) What is $\EX{\S_n}$? 
  \item (15 points) Upper-bound the following probability
    $$\probX{\S_n-\EX{\S_n}\geq E} $$
  \end{itemize}

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

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%%%%%%%%%%%% PROBLEM 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries Coin-tossing: Word Problem.} (20 points)
Suppose you have access to a coin that outputs heads with probability $1/2$ and outputs tails with probability $1/2$. 
Let $\S_n$ represent the {\em number of coin tosses needed} to see exactly $n$ heads. 
\begin{itemize}
\item (5 points) What is $\EX{\S_n}$? 
\item (15 points) Upper-bound the following probability
  $$\EX{\S_n-\EX{\S_n} \geq E} $$
\end{itemize}

 


\noindent{\bfseries Solution.}\newline 
\   %%% <-- ERASE THIS LINE AND WRITE YOUR SOLUTION HERE
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%%%%%%%%%%%% PROBLEM 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries Sum of Poisson.} 
  (25 points) Let $\Y$ be the random variable over sample space $\{0,1,2,\dots\}$ such that $\Pr[\Y=k]=\frac{e^{-\mu}\mu^k}{k!}$. 
  This is the Poisson distribution with parameter $\mu$.
  \begin{itemize}
      \item (3 points) Prove that the mean of a Poisson distribution with parameter $\mu$ is equal to $\mu$.
      \item (7 points) Prove that if $\Y_1$ and $\Y_2$ are independent Poisson distributions with parameters $\mu_1$ and $\mu_2$ respectively, then the random variable $\Y_1+\Y_2$ is also a Poisson distribution with parameter $\mu_1+\mu_2$.
      \item (15 points) Let $\X$ be the Poisson distribution with mean $m/n$. 
  Let $\S_n \defeq \X\p1+\X\p2+\dotsi+\X\p n$, where $\X\p1,\X\p2,\dotsc,\X\p n$ are all independent and identical to $\X$. 
  Upper-bound the following probability
    $$\probX{\S_n - \EX{\S_n} \geq E} $$
  \end{itemize}

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


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%%%%%%%%%%%% PROBLEM 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries Empty Bins in the Poisson Model.} (20 points)
Let $\X$ represent the Poisson distribution with mean $m/n$. 
Let $\Y$ be the indicator variable $\1{\X=0}$. 
That is, $\Y$ is the random variable that is $1$ if and only if the random variable $\X$ is 0.

Let $\S_n = \Y\p1+\Y\p2+\dotsc+\Y\p n$, where $\Y\p1,\Y\p2,\dotsc,\Y\p n$ are independent and identical to $\Y$. 
\begin{itemize}
\item (5 points) What is $\EX{\S_n}$? 
\item (15 points) Upper-bound the following probability
  $$ \probX{\S_n-\EX{\S_n} \geq E}$$
\end{itemize}

\noindent{\bfseries Solution.}\newline 
\newpage



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%%%%%%%%%%%% PROBLEM 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries Another proof for Chernoff bound} (15 points)
Consider the following simple type of Chernoff Bound:\\

Suppose $\S_n=\sum_{i=1}^n \X\p i$ where $\X\p1,\X\p2,\dots,\X\p n$ are i.i.d Bernoulli random variables such that, $\X=\bern{p}$. Then, for any $\eps>0$, the following Chernoff bound states:
$$\Pr[\S_n\geq n(p+\eps)]\leq e^{-n\kl{p+\eps}{p}}$$
To prove the inequality above, we define i.i.d Bernoulli random variables ${\X'}\p1,{\X'}\p2,\dots,{\X'}\p n$ such that ${\X'}=\bern{p+\eps}$. 
Define $\S'_n\defeq \sum_{i=1}^{n}{\X'}\p i$.
\begin{itemize}
    \item (3 points) Define $h_k\defeq\frac{\Pr[\S'_n=k]}{\Pr[\S_n=k]}$ and obtain a simplified expression for $h_k$.
    \item (7 points) For any $k\geq n(p+\eps)$, prove that $h_k\geq e^{n\kl{p+\eps}{p}}$ .
    \item (5 points) Use the inequality above to prove the Chernoff bound $$\Pr[\S_n\geq n(p+\eps)]\leq e^{-n\kl{p+\eps}{p}}.$$
\end{itemize}
\noindent{\bfseries Solution.}\newline 
\newpage

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%%%%%%%%%%%% PROBLEM 6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries Random Walk in 2-D.} (20 points)
Suppose an insect starts at $(0,0)$ at time $t=0$. 
At time $t$, its position is described by $(\X(t),\Y(t))$. 
At the next time step $t+1$, the insect uniformly at random moves to (a) $(\X(t)+1,\Y(t))$, $(\X(t)-1,\Y(t))$, $(\X(t),\Y(t)+1)$, or $(\X(t),\Y(t)-1)$. 

State (5 points) and prove (15 points) a theorem that bounds how far from the origin the insect is at time $t=n$.  

\noindent{\bfseries Solution.}\newline




\end{enumerate}
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%%%%%%%%%%%% PLEASE LIST COLLABORATORS BELOW  %%%%%
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{\bfseries Collaborators :} \newline 
% ENTER THEIR NAMES HERE  

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