Question: 2016 Fall Final - 23
Consider the set $S = \{1,2,3,...,10\}$. You choose a uniformly random element $z$ in $S$.
Define the random variables
$$
X =
\begin{cases}
0\; \ \text{if \(z\) is even}, \\
1\; \ \text{if \(z\) is odd}
\end{cases}
$$
and
$$
Y =
\begin{cases}
0\; \ \mathrm{if}\ z \in \{1,2\}, \\
1\; \ \mathrm{if}\ z \in \{3,4,5,6\}, \\
2\; \ \mathrm{if}\ z \in \{7,8,9,10\}.
\end{cases}
$$
Which of the following is true?
(a)
The random variables $X$ and $Y$ are independent.
(b)
None of the above.
(c)
The random variables $X$ and $Y$ are not independent.