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\rhead{Name: Hemanta K. Maji} %%% <-- YOUR NAME HERE

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\title{Homework 4\\(50 points)}

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%%% Problem 1 
\item {\bfseries (15 + 5 points) Most Functions are Small Biased.}
  Let $f\colon\zo^n \to \{+1,-1\}$ be a boolean function. 
  Our objective is to consider a \ul{random} boolean function $f$. 
  A random boolean function (the distribution is represented by $\F_n$) is generated as follows. 
  
  For every input $x\in\zo^n$, choose the value of $f(x)$ independently to be $+1$ with probability $1/2$, and $-1$ with probability $1/2$. 
  
  
  \begin{enumerate}
  \item Fix $S\in \zo^n$. 
    For every $x\in\zo^n$, note that $f(x)\cdot\chi_S(x)$ is independently $+1$ with probability $1/2$, and $-1$ with probability $1/2$. 
    Using this observation, upper-bound the following probability
    $$\probX{\hf(S)\geq\eps \colon f\drawn \F_n}$$
  \item Use the previous result to lower-bound the following probability
    $$\probX{\forall S\in\zo^n , \hf(S)\leq \eps \colon f\drawn \F_n}$$
  \end{enumerate} 
  
  
    
 
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%%% Problem 2
\item {\bfseries (10 + 20 points) Monotonicity of Norm.} 
  Define the following function
   $$f(p) = \frac1p \log\left(\frac1n\sum_{i=1}^n a_i^p\right)$$  
  Note that in this function we have \ul{fixed} the values of $a_1,\dotsc,a_n$. 
  \begin{enumerate}
  \item Calculate $\frac{\dif f}{\dif p}$. 
  \item Use Jensen's inequality to prove that $\frac{\dif f}{\dif p} \geq 0$, for $p\geq 1$. 
    And equality holds if and only if $a_1=a_2=\dotsi=a_n$. 
  \end{enumerate} 
  This proves that the norm is non-decreasing, and equality holds if and only if all $a_i$s are equal! 
  
  

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