

\subsection{\bf Hypothesis Testing---II}

\begin{slide}
\begin{itemize}
\item We are given our Standard $N$ observations $(x_1 \ldots x_N)$ on
which to make our decision.

\item We do this by constructing yet another statistic $Y$ which is
some function of $x_1 \ldots x_N$.

\item We assume that $Y$ has distribution $P_Y^{\ (i)}$ if hypothesis
$i$ is true ($i=0$ or 1 if two hypotheses).

\item $Y$ could be vector valued but we will take scalar case here.

\item In the best of all possible worlds, one can find a $Y$ such that
the values taken by it under two hypotheses are disjoint, i.e.,
\begin{equation}
\int\limits P^{\ (0)}_Y (y) P^{\ (1)}_Y (y) dy = 0
\end{equation}
\item Unfortunately, Equation~(36) is not always attainable 

\begin{description}
\item{$\bullet$} but one should always try to minimize the probability
overlap integral.
\end{description}
\end{itemize}
\end{slide}

