Following beamer syntax generates an error:
\begin{frame}
\frametitle{Sys}
$$Y \sim \left\{\begin{matrix}
\chi_{2N}^2,~~~~~~~~~:H_0\\
\chi_{2N}^2(2\gamma),~~~:H_1
\end{matrix}\right$$
$$f_Y(y) = \left\{\begin{matrix}
\frac{1}{2^N\Gamma(N)}y^{N-1}e^{\frac{-y}{2}},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:H_0\\
\frac{1}{2}\left ( \frac{y}{2N\gamma}\right)^\frac{N-1}{2}e^{\left(-\frac{y+2N\gamma}{2}\right)}I_{N-1}\left ( \sqrt{2N\gamma y} \right),:H_1 \end{matrix}\right$$
\end{frame}

\right.(you forget the period.) at the end of each math equation. – Jesse May 18 '14 at 06:39$$...$$, it's highly deprecated.Use the
– MaxNoe Aug 16 '14 at 10:22equationoralignenvironments from the amsmath package.equationfor single-lign equations,alignfor multi-lign equations. You can use the &-sign do the aligning inalign. Have a look to this please: ftp://ftp.dante.de/tex-archive/info/l2tabu/english/l2tabuen.pdf