<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\begin{document}
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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 $j\
(i&lt;j)$.

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

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


\vspace{.25in}

{\bf Answer}

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

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

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)]
$\left(\begin{array}{cccc} 1 &amp; -0.534 &amp; -0.408 &amp; -0.272\\ -0.534 &amp;
1 &amp; -0.327 &amp; -0.218\\ -0.408 &amp; -0.327 &amp; 1 &amp; -0.167\\ -0.272 &amp;
-0.218 &amp; -0.167 &amp; 1 \end{array} \right)$ = correlation matrix

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

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


\end{document}
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