\problem {\bf (12 points)} 

Let random variables $X$ and $Y$ be the outcomes of two dice rolls.
Assume that the dice are fair, 6-sided, and independent.  Each problem
part below lists two random variables derived from $X$ and $Y$.  State
whether or not these two random variables are independent and justify
your answer.

\bparts
\ppart  $(X \bmod 3) + (Y \bmod 3)$ and $(X \bmod 2) + (Y \bmod 2)$

\solution{Independent.  $X \bmod 3$ and $X \bmod 2$ are independent,
which you can see by drawing a 3x2 table of values of $X \bmod 3$
versus values of $X \bmod 2$.  Obviously, $Y\bmod 3$ and $Y \bmod 2$
are independent as well.  Furthermore, it is true in general that if
$A$ is independent of $B$ and $A'$ is independent of $B'$ then
$f(A,B)$ is independent of $g(A',B')$ for any functions $f$ and $g$.
Here the two functions are addition over integers.}


\ppart $((X+Y)\bmod{2})$ and $((X*Y)\bmod{2})$

\solution{ These random variables are not independent.  $(X+Y\bmod 2)
= 1$ iff $X$ and $Y$ have different parity.  $(X*Y\bmod 2) =1$ iff
both $X$ and $Y$ have odd parity.  Therefore, if $(X*Y\bmod 2) =1$
then $(X+Y\bmod2)=0$, and hence the two events depend on each other.}

\eparts