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\begin{document}
\handout{quiz3}{Quiz 3}{May 7, 1996}

\begin{problems}

\problem
Let $X$ be a $\{1,0,-1\}$ valued random variable and $Y$ a
$\{1,-1\}$ random variable with joint distribution given by the following
table:

\[
\begin{array}{|r|r|c|} \hline
j     & k        & P((X=j) \wedge (Y=k))\\ \hline\hline
1 & 1 & a \\ \hline
0 & 1 & b \\ \hline
-1 & 1 & c \\ \hline
1 & -1 & d \\ \hline
0 & -1 & e \\ \hline
-1 & -1 & f \\ \hline
\end{array}
\]
For example, $P((X=0) \wedge (Y=1))= b$.

For each of the quantities in the box on the left below, indicate the
number of the expression for it from the box on the right.  For example,
the first entry is filled in with the answer ``(14)''.\\
$\begin{array}{|c|c|}\hline
P((X=0) \wedge (Y=1)) & \mbox{(14)} \\ \hline
P(X=0) & \\ \hline
P(Y=-1) & \\ \hline
P((Y=1) \mid (X=1)) & \\ \hline
P((Y=1) \mid (X\geq 0)) & \\ \hline
E(X) & \\ \hline
E(Y) & \\ \hline
E(X^{2}) & \\ \hline
\mbox{Var}(X) & \\ \hline
E(X\cdot Y) & \\ \hline
\mbox{Cov}(X,Y) & \\ \hline
\end{array}$\qquad\small{\begin{tabular}{|l|}\hline
(1) $a$\\
(2) $a+b$\\
(3) $b+e$\\
(4) $d+e+f$\\
(5) $a+b+d+e$\\
(6) $a+d-c-f$\\
(7) $a+c+d+f$\\
%or 1-b-e
(8) $a+f-c-d$\\
(9) $2(a+b+c)-1$\\
% or a+b+c-d-e-f or 1-2(d+e+f)
(10) $a/(a+d)$\\
(11) $(a+b)/(a+b+d+e)$\\
% or (a+b)/(1-c-f)
(12) $a+c+d+f-(a+d-c-f)(1-b-2c-e-2f)$\\
% or 1-b-e-(a+d-c-f)^{2}
(13) $a+f-c-d-(a+d-c-f)(a+b+c-d-e-f)$\\
% or a+f-c-d-(a+d-c-f)(1-2(d+e+f))
(14) $b$\\ \hline
\end{tabular}}
\end{problems}
\end{document}
