

\subsection{\bf Robust Determination of Mean---III}

\begin{slide}
\begin{itemize}
\item For a trivial example, suppose I measured $y=x^2$ where $x$ had
an exponential distribution.

\item Then the best estimate of the mean of $y$ is obviously
\[
y^*_{\rm clever} = \left( \frac{1}{N} \sum^N_{i=1} \left( y^{(i)}
\right)^{1/2} \right)^2
\]
and {\em not}
\[
y^*_{Am} = \frac{1}{N} \sum^N_{i=1} y^{(i)}
\]

\item The problem of estimating $\mu$ is hardest for distributions
$p(x)$ that have long tails.  

\item Then with a small probability, we will
generate events from the tail, but these will be way off the means,

\begin{description}
\item{$\bullet$} and if included in calculation of mean, will distort
results in a finite sample.
\end{description}
\end{itemize}
\end{slide}

