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Name \underline{\hspace{60mm}} \\ 
$\,$ \\
Student Number \underline{\hspace{60mm}}
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\begin{center}   
{\Large \textbf{STA 431 Quiz 1}}\\
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\noindent
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\begin{enumerate} 

\item (5 points) 
Let the random variable $x$ have expected value $\mu_x$, let the random variable $y$ have expected value $\mu_y$, and let $a$ be a non-zero constant.
Circle one the of following statements and prove it, \emph{using the definition of covariance} from the formula sheet. 

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$Cov(ax,y) = a^2Cov(x,y)$, & $Cov(ax,y) = aCov(x,y)$, & $Cov(ax,y) = Cov(x,y)$,   & $Cov(ax,y) = 0$
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\newpage

\item (5 points) Let 
\begin{eqnarray*}
    y_1 & = & \alpha_1 + \beta_1 x + \epsilon_1 \\
    y_2 & = & \alpha_2 + \beta_2 x + \epsilon_2,
\end{eqnarray*}
where $E(x) = \mu$, $Var(x) = \sigma^2_x$, $E(\epsilon_1) = E(\epsilon_2) = 0$, $Var(\epsilon_1) = \sigma^2_1$ and $Var(\epsilon_2) = \sigma^2_2$. The random variables $x$, $\epsilon_1$ and $\epsilon_2$ are independent. Using anything you wish from the formula sheet, calculate $Cov(y_1,y_2)$. \textbf{Circle your final answer.}


\end{enumerate}


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