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\title{CS761 Spring 2015 Homework 1}
\author{Assigned Jan. 18, due Jan. 20 before class}
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\begin{document}
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Instructions: 
\begin{itemize}
\item Homeworks are to be done individually.
\item Typeset your homework in latex using this file as template (e.g. use pdflatex).  Show your derivations.
\item Hand in the compiled pdf (not the latex file) online.  Instructions will be provided.  We do not accept hand-written homeworks.  
\item Homework will no longer be accepted once the lecture starts.
\item Fill in your name and email below.  
\end{itemize}

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Name:                      

Email: 

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Each question has 5 points if answered correctly.
Wrong answers or blank answers 0 points.
However, an explicit ``I don't know'' answer is worth 1 point.


\begin{enumerate}

\item Define inner product on real functions
$$\langle g, h \rangle = \int_0^1 g(x)h(x)\,dx.$$
Orthogonalize the basis $1, x, x^2$.

\item Here is a little fact: Let $X$ be a random variable with finite mean $\mu$ and finite variance $\sigma^2$.  Then $\forall t>0$ we have
$$P(|X-\mu| \ge t) \le \frac{\sigma^2}{t^2}.$$
Now let $X_1, X_2, \ldots$ be a sequence of independent and identically distributed random variables with mean $\mu$ and variance $\sigma^2$.
Let $\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$.
Use the little fact to prove 
$$\forall \epsilon>0, \forall \delta>0, \exists N>0, \forall n\ge N, P(|\bar{X}_n - \mu|\le \epsilon)\ge 1-\delta.$$


\end{enumerate}

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