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\title{Problem Set 9}
\author{Randomized Algorithms}
\date{Due Friday, November 8}
\begin{document}
\maketitle

\begin{enumerate}
\item
 Let $X_1$ and $X_2$ be zero-mean subgaussians with parameters
 $\sigma_1$ and $\sigma_2$, respectively.
 \begin{enumerate}
 \item Show that if $X_1$ and $X_2$ are independent, then $X_1X_2$
   is subgamma.  What are the parameters in terms of $\sigma_1$ and
   $\sigma_2$?
 \item Show that in general, without assuming independence,
   $X_1+X_2$ is subgaussians with parameter
   $2\sqrt{\sigma_1^2+\sigma_2^2}$.
 \end{enumerate}
% \item Suppose you can sample elements $x_1, \dots, x_m$ from a
%   mean zero subgaussian variable $X$, and would like to compute its
%   \emph{kurtosis}
%   \[
%     \kappa = \frac{\E[X^4]}{\E[X^2]^2}.
%   \]
%   Suppose that the variance, subgaussian parameter, and kurtosis are
%   all constants.

%   \begin{enumerate}
%   \item Consider the algorithm that computes the empirical fourth and
%     second moments $\frac{\sum x_i^4}{m}$ and $\frac{\sum x_i^2}{m}$,
%     and outputs the former divided by the latter squared.  How large
%     must $m$ be for this to be an $(\eps, \delta)$ approximation to
%     $\kappa$?
%   \item Construct a different algorithm that achieves an $(\eps,
%     \delta)$ approximation to $\kappa$ with fewer measurements. 
%   \end{enumerate}
\end{enumerate}

\end{document}
