Q:

What is the difference between the joint pdf of (X,Y) and the pdf of XY?

Let X and Y be dependent random variables. I know that the joint pdf of (X,Y) can be represented as a table. For example, consider the following representation of the pdf of (X,Y):
$\left[\begin{array}{cccc}
 & 0 & 1 & 2       \\
0 & 1/16 & 1/16 & 2/16       \\
1 & 1/16 & 6/16 & 1/16       \\
2 & 2/16 & 1/16 & 1/16
\end{array}\right]$ 
In this case, imagine that the $[0,1,2]$ in the first row represent the possible values of Y and the $[0,1,2]$ in the first column represents the possible values of X. Then, the other values represent the the probability of a given combination of X and Y. 
But what is the pdf of XY? Can it also be represented as a table?
Thanks

A:

It might help to remember that $X$ is a random variable taking values $\{0,1,2\}$ and that $Y$ is a random variable taking values $\{0,1,2\}$.
Then, the random variable $Z = XY$ is a random variable which can take the values $\{0,1,2,4\}$.
Notice that $Z = 4$ only if $X = 2$ and $Y = 2$, which happens with probability $1/16$. You can fill out the distribution of $Z$ by continuing where I left off for the other values of $Z$.

