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Prof.~David Draper \\
Department of Statistics \\
University of California, Santa Cruz
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\textbf{\large AMS 131: Quiz 7}
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Name: \underline{\hspace*{5.85in}}
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All parts of this problem are unrelated (i.e., the assumptions in part (x) apply only to part (x)). All expectations and variances are assumed to exist and to be finite.
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\item[(a)]
You're working with two random variables $X$ and $Y$, which may be dependent and for which $V ( X ) = V ( Y )$. Show that the random variables $W = ( X + Y )$ and $Z = ( X - Y )$ are uncorrelated. \textit{Hint:} Nothing fancy --- just simplify the covariance of $W$ and $Z$, using properties of covariance discussed in class and discussion section.
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\item[(b)]
You're working with two random variables $X$ and $Y$ that are negatively correlated. Which is bigger --- $V ( X + Y )$ or $V ( X - Y )$ --- or are they equal? Show your calculations.
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\item[(c)]
You're working with two random variables $X$ and $Y$ such that $V ( X ) = 9, V ( Y ) = 4$, and $\rho ( X, Y ) = - \frac{ 1 }{ 6 }$. Compute $V ( X + Y )$ and $V ( X - Y )$ (show your calculations).
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\item[(d)]
You and your research assistant (RA) are working with two random variables $X$ and $Y$, and your RA has computed the following values: $E ( X ) = 3, E ( Y ) = 2, E ( X^2 ) = 10, E ( Y^2 ) = 29$, and $E ( X \, Y ) = 0$. Show that there must be something wrong in this computation. \textit{Hint:} Consider the bounds on variances and correlations.
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