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{\bf Question}

Suppose that $40\%$ of the students in a large population are
freshman, $30\%$ are sophomores, $20\%$ are juniors and $10\%$ are
seniors.  Suppose that 10 students are selected at random from te
population; and let $X_1,X_2,X_3,X_4$ denote, respectively, the
numbers of freshmen, sophomores, juniors, and seniors that are
obtained.

\begin{description}
\item[(a)]
Determine $\rho(X_i,X_j)$ for each pair of values $i$ and $j\
(i<j)$.

\item[(b)]
For what values of $i$ and $j\ (i<j)$ is $\rho(X_i,X_j)$ most
negative?

\item[(c)]
For what values of $i$ and $j\ (i<j)$ is $\rho(X_i,X_j)$ closest
to 0?
\end{description}


\vspace{.25in}

{\bf Answer}

\begin{tabular}{r|l|c|}  {} & Prob. & Number\\ \hline Freshman & $p_1=0.4$
& $x_1$\\ Sophomore & $p_2=0.3$ & $x_2$\\ Junior & $p_3=0.2$ &
$x_3$\\ Senior & $p_4=0.1$ & $x_4$\\ \hline \end{tabular}

n=10. ${\rm var}(X_i)=np_i(1-p_i)$\ \ \ ${\rm
cov}(X_i,X_j)=-np_ip_j$

${\rm var}(X_1)=2.4,\ \ {\rm var}(X_2)=2.1,\ \ {\rm
var}(X_3)=1.6,\ \ {\rm var}(X_4)=0.9$

We can form a matrix of variances and covariances.

$$\bordermatrix {    & X_1  & X_2 & X_3 & X_4 \cr
                 X_1 & 2.4  & -10(0.4)(0.3) & -10(0.4)(0.2) & -10(0.4)(0.1) \cr
                 X_2 & -1.2 & 2.1 & -10(0.3)(0.2) & -10(0.3)(0.1) \cr
                 X_3 & -0.8 & -0.6 & 1.6 & -10(0.2)(0.1) \cr
                 X_3 & -0.4 & -0.3 & -0.2 & 0.9 \cr} $$

Then we can find the correlations $\ds \rho_{X_i,X_j}=\frac{ {\rm
cov}\left(X_i,X_j \right)}{\sqrt{{\rm var}\left(X_i\right){\rm
var}\left(X_j\right)}}$
\begin{description}
\item[(a)]
$\left(\begin{array}{cccc} 1 & -0.534 & -0.408 & -0.272\\ -0.534 &
1 & -0.327 & -0.218\\ -0.408 & -0.327 & 1 & -0.167\\ -0.272 &
-0.218 & -0.167 & 1 \end{array} \right)$ = correlation matrix

\item[(b)]
$i=1,\ j=2$

\item[(c)]
$i=3,\ j=4$
\end{description}


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