<s>[INST] Random Variables, Die toss A fair die is tossed twice. Let $X$ = the ssm of the faces, $Y$= the maximum of the two faces, and $Z$=|face 1 - face 2|.
write down the value of $X,Y,$ and $W=XZ$ for each outcome $w\in\ S$
I already found the value and range of $X,Y$ but I'm not sure how to find $W=XZ$.
I saw someone post a similar question already answered however, it wasn't explained how to find $W=XZ$.
 [/INST] Since you said you've enumerated the outcomes for $X$, do the same for $Z$.  Below I made a table for the values of both $X$ and $Z$.  Can you now make the corresponding table for $W = XZ$?
$$
\begin{array}{|c|c|c|c|c|c|c|}
\hline
X& 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 \\
2 & 3 & 4 & 5 & 6 & 7 & 8 \\
3 & 4 & 5 & 6 & 7 & 8 & 9 \\
4 & 5 & 6 & 7 & 8 & 9 & 10 \\
5 & 6 & 7 & 8 & 9 & 10 & 11 \\
6 & 7 & 8 & 9 & 10 & 11 & 12\\
\hline
\end{array}
$$
$$
\begin{array}{|c|c|c|c|c|c|c|}
\hline
Z & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
1 & 0 & 1 & 2 & 3 & 4 & 5 \\
2 & 1 & 0 & 1 & 2 & 3 & 4 \\
3 & 2 & 1 & 0 & 1 & 2 & 3 \\
4 & 3 & 2 & 1 & 0 & 1 & 2 \\
5 & 4 & 3 & 2 & 1 & 0 & 1 \\
6 & 5 & 4 & 3 & 2 & 1 & 0 \\
\hline
\end{array}
$$
</s>