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COMPUTER SCIENCE 20, SPRING 2012 \\
DISCRETE MATHEMATICS FOR COMPUTER SCIENCE\\
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Class \#28 (Statistics)
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\paragraph*{Homework, due in hard copy Wednesday 4/18/2012 at 10:10am}
\begin{enumerate}
\item Your financial advisor tells you about two possible investment options for your \$1000. The first option is investing in Apple shares. The Apple shares have a 80\% chance of doubling in value and a 20\% chance of losing all of their value. The second option is called a \emph{principal guaranteed} purchase of the Apple shares. In this scenario, even if the shares lose their value you still get back the original money you invested in Apple. However, the guarantee costs a \textbf{non-refundable} fee, $x$, so that the amount you can invest in the Apple shares is only \[\$1000-x\]
\begin{enumerate}
\item What is the value of $x$ so that the two options have the same expected return?
\item Assume the fee is the value you found above, so that the two options have the same expected return. Which of the two options has a higher variance on the return?\footnote{People who would choose the option with the lower variance are called \emph{risk-averse}. People who would choose the option with the higher variance are called \emph{risk-seeking}. People who are neutral are called \emph{risk-neutral}.}
\end{enumerate}
\item Suppose a particular subset of CS 20 students had the following scores for the first two homework assignments: \[ \langle(0,3),(6,6),(4,5),(7,8),(8,8)\rangle\] That is, some student scored 0 on the first homework and 3 on the second homework; another scored 6 on the first homework and 6 on the second homework, etc.
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\item Compute the following quantities for the two homework assignments, \emph{separately}. For example, compute the sample mean of the scores on the \emph{first} homework and then the mean of the scores on the \emph{second} homework, and write them both down where it asks for the mean.
\begin{enumerate}
\item mean
\item variance
\item standard deviation (to 2 decimal places)
\item median
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\item So far we have only computed quantities relating to the two homeworks separately. To gain some insight into how they vary together, compute the sample covariance and correlation coefficient (to 2 decimal places) of the score on the first homework with the score on the second homework. Provide an interpretation of the correlation coefficient (one sentence only, please!).
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