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\lhead{CS 590--MTK, SPRING 2019}               
\rhead{Name: Hemanta K. Maji} %%% <-- REPLACE Hemanta K. Maji WITH YOUR NAME HERE

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\title{Homework 2}

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%%%%%%%%%%%% PROBLEM 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries Solving an Interesting Equation.} (20 points)
Our objective is to understand the behavior of $x$ such that $x! = n$ as a function of $n$. 
We shall use the following estimate of $x!$ 
  $$ \left(\frac x\me\right)^x ~\leq~ x! ~\leq~ x^x$$
(Remark: The upper-bound is trivial. The lower-bound is a consequence of estimating the increasing function $\log t$ using integrals.)
\begin{itemize}
\item Prove that if $x=\frac{\log n}{\log\log n}$ then $x! \leq n$. 
\item Prove that, for large enough $n$, if $x=\frac{\me\log n}{\log\log n}$ then $x!\geq n$. 
\end{itemize}
(Remark: These proofs complete the argument that $x=\Theta(\log n/\log\log n)$. 
Substituting $\poly n$ instead of $n$, completes the argument that $x=\Theta(\log n/\log\log n)$ when $x!=\poly n$, for any fixed polynomial $\poly$) 


\noindent{\bfseries Solution.}\newline 
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%%%%%%%%%%%% PROBLEM 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries Upper-bounding Max-load using Poisson Approximation Theorem.} (20 points)
Recall that in the lecture we proved the upper-bound on max-load directly. 
Let us see how we can use the Poisson approximation theorem to prove that result easily. 

\begin{itemize}
\item Let $\X(\mu)$ be the Poisson distribution with mean $\mu$.   Prove the following bound. 
  For any integer $T\geq 2\mu$, we have 
  $$\probX{\X(\mu) \geq T} \leq 2\probX{\X(\mu)=T}$$
  (Remark: Basically, this inequality proves that $\probX{\X(\mu) \geq T}$ is well approximated by $\probX{\X(\mu)=T}$)
\item Suppose $\X$ represents the Poisson distribution with mean $\mu=1$. 
  Prove that there exists a positive constant $c$ such that
    $$\probX{\X \geq c\frac{\log n}{\log\log n}} \leq \frac1{n^3} $$
\item Prove that 
  $$ \probX{\max\left\{\X\p1,\X\p2,\dotsc,\X\p n\right\} < c\frac{\log n}{\log\log n}} \geq 1-\frac1{n^2}$$
\end{itemize} 

\noindent{\bfseries Solution.}\newline 
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%%%%%%%%%%%% PROBLEM 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries Coupon Collector Problem.} (20 points)
Our objective is to solve the Coupon Collector Problem using the Poisson approximation theorem. 
Here, we want to determine the value of $m$ such that when $m$ balls are thrown into $n$ bins, with high probability every bin receives at least one ball. 
Equivalently, we want to determine the value of $m$ such that the probability of the minimum load being $0$ is small. 

 
\begin{itemize}
\item Let $\X(\mu)$ be the Poisson distribution with mean $\mu$.  
Find the value of $m$ such that, for $\mu=m/n$, we have
  $$\probX{\X(\mu)=0} \leq \frac1{n^3}$$
\item Let $\X$ be the Poisson distribution for the $\mu$ determined above. 
  Let $\X\p i$ be the $i$-th independent copy of the distribution $\X$. 
  Prove the following bound. 
  $$ \probX{\min\left\{\X\p1,\X\p2,\dotsc,\X\p n\right\} =0} \leq \frac1{n^2}$$
\item Use the Poisson Approximation Theorem to prove the following bound.
  For large enough $n$ and   $\rvL_{\min}\defeq\min\left\{\rvL_1,\rvL_2,\dotsc,\rvL_n\right\}$
  $$\probX{\rvL_{\min}\geq 1} \geq 1-\frac1n$$
\end{itemize}

\noindent{\bfseries Solution.}\newline 
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\item {\bfseries A Fun Ball and Bins Problem.} (20 points)
Let us consider a fun problem. 
Suppose we are interested in ensuring that every bin received at least $2$ balls. 
Let us get you started on how to think about this fun problem. 

Let $\X(\mu)$ be the Poisson distribution with mean $\mu$. 
Find the value of $m$ such that, for $\mu=m/n$ and positive integers $m$ and $n$, we have
  $$ \probX{\X(\mu) \in\{0,1\}} \leq \frac1{n^3} $$

\noindent{\bfseries Solution.}\newline 
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%%%%%%%%%%%% PROBLEM 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries Towards proving Poisson Approximation Theorem.} (20 points)
Let me get you started towards proving the simpler version of the Poisson approximation theorem that was taught in the class. 
Let $\X$ be the Poisson distribution with mean $\mu$, where $\mu = m/n$. 
Prove the following inequality
  $$\probX{\X\p1 + \X\p2 + \dotsi + \X\p n = m}  = \left(\frac m \me\right)^m\frac1{m!} \geq  \frac1{\me\sqrt m}$$
(Remark: Use the Stirling's approximation taught in the class for the final inequality.)

\noindent{\bfseries Solution.}\newline 
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%%%%%%%%%%%% PLEASE LIST COLLABORATORS BELOW  %%%%%
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{\bfseries Collaborators :} \newline 
% ENTER THEIR NAMES HERE  

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