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% correct bad hyphenation here
\hyphenation{op-tical net-works semi-conduc-tor}

\begin{document}

\begin{itemize}
\item \textbf{a)}
\\
\\
\begin{tabular}{ r l l r l}
$\hat{y}_{o}$ & $=x_{o}^{'}~\hat{\beta}$ & & $\hat{y}_{o}$ & $=x_{o}^{'}~\hat{\beta}+x_{o}^{'}\left( \hat{\beta} - \beta  \right) $ \\
& $=x_{o}^{'}\left( X^{'}X \right)^{-1}X^{'}Y$ & 
   & & $=x_{o}^{'}~\beta + x_{o}^{'}\left( \left( X^{'}X \right)^{-1}X^{'}Y - \beta  \right) $ \\
& $=x_{o}^{'}\left( X^{'}X \right)^{-1}X^{'} \left( X\beta + \epsilon \right)$ & 
   & & $=x_{o}^{'}~\beta + x_{o}^{'}\left( \left( X^{'}X \right)^{-1}X^{'}\left( X\beta + \epsilon \right) - \beta  \right) $ \\
& $=x_{o}^{'} \underbrace{\left( X^{'}X \right)^{-1}X^{'}X}_{I}  \beta + x_{o}^{'} \left( X^{'}X\right)^{-1}X^{'}\epsilon$ &
   & & $=x_{o}^{'}~\beta + x_{o}^{'}\left( \underbrace{\left( X^{'}X \right)^{-1}X^{'}X}_{I}\beta + \left( X^{'}X\right)^{-1}X^{'}\epsilon - \beta  \right) $ \\
& $=x_{o}^{'}\beta+\sum_{i=1}^{n}{l_{i} \left(x_{o} \right) } \epsilon_{i}$ & 
   & & $=x_{o}^{'}~\beta + x_{o}^{'}\left( \beta + \left( X^{'}X \right)^{-1}X^{'}\epsilon -\beta \right)$ \\
 & & & & $=x_{o}^{'}~\beta + x_{o}^{'}\left( X^{'}X \right)^{-1}X^{'}\epsilon$ \\
 & & & & $=x_{o}^{'}\beta+\sum_{i=1}^{m}{l_{i} \left(x_{o} \right) } \epsilon_{i}$ \\
 \end{tabular}
 \\
 \\
 
\item
\textbf{b)}
\\
\\
\begin{tabular}{ r l}
$EPE \left( x_{o} \right) $ & $=E_{x_{o}} \left[ E_{y_{o}/x_{o}} \left( y_{o} - f(x_{o}) \right)^{2} \right] $ \\
 & $=E_{\tau} \left[ E_{y_{o}/x_{o}} \left( y_{o} - x_{o}^{'}\beta \right)^{2} \right] $ \\
  & $=E_{\tau} \left[ E_{y_{o}/x_{o}} \left( y_{o}^{2} - x_{o}^{'}\beta\beta^{'}x_{o} - 2y_{o}x_{o}\beta \right) \right] $ \\
  & $=E_{\tau} \left[ V_{y_{o}/x_{o}}(y_{o}) + x_{o}^{'}V_{y_{o}/x_{o}}(\beta)x_{o}^{'} - 2Cov(y_{o},x_{o}\beta) \right] $~, where $V(\beta)=(X^{'}X)^{-1}\sigma^2$ \\
 & $=E_{\tau} \left[ \sigma^2 + x_{o}^{'} \left( X^{'}X \right)^{-1}x_{o}\sigma^2 - 0 \right] $ \\
 & $=\sigma^2 + E_{\tau} \left[x_{o}^{'} \left( X^{'}X \right)^{-1}x_{o} \right]  \sigma^2 $ \\
\end{tabular}
\\
\\

\item
\textbf{c)}
\\
\\
\begin{tabular}{l l}

$EPE \left( x_{o} \right) $ & $ = \sigma^2 + E_{\tau} \left( x_{o}^{'}(X^{'}X)^{-1}x_{o} \right)\sigma^2 $\\
& $ = \sigma^2 + E_{\tau} \left( x_{o}^{'}~Cov(X)^{-1}~\frac{x_{o}}{N} \right)\sigma^2 $\\
& $ = \sigma^2 + \frac{\sigma^2}{N}~E_{\tau} \left( x_{o}^{'}~Cov(X)^{-1}x_{o} \right) $\\
& $ = \sigma^2 + \frac{\sigma^2}{N}~tr\left( x_{o}^{'}~Cov(X)^{-1}x_{o} \right) $\\
& $ = \sigma^2 + \frac{\sigma^2}{N}~tr\left( x_{o}x_{o}^{'}~Cov(X)^{-1} \right) $\\
& $ = \sigma^2 + \frac{\sigma^2}{N}~tr\left( Cov(X)~Cov(X)^{-1} \right) $\\
& $ = \sigma^2 + \frac{\sigma^2}{N}P $\\
& $ = \sigma^2 + \left( \frac{P+N}{N} \right) $\\
\end{tabular}

 
\end{itemize}

\end{document}