What is the joint distribution of random variables Y and XY? Random variables $X$ and $Y$ are independent. $X$ takes on values of $-1$ and $1$ with $\frac{1}{2}$ probability each, $Y$ takes on values of $-1,0,1$ with $\frac{1}{3}$ probability each. What is the joint distribution of random variables $Y$ and $XY$?
I've calculated the mere distribution of $X\cdot Y$, but I'm confused with the wording "joint distribution of $Y$ and $XY$". What does it mean and what is actually required to do?
 A: Hint:
Let $f(x,y)\ge 0$. Compute
$$
Ef(XY, Y)
$$

this is
$$
\frac 16\left[
f(-1 \times -1, -1) + f(-1 \times 0, 0) + f(-1 \times 1, 1) +\\
f(1 \times -1, -1) + f(1 \times 0, 0) + f(1 \times 1, 1) 
\right] \\=
\frac 16\left[
f(1, -1) + 2f(0, 0) + f(-1, 1) +
f(-1, -1)  + f(1, 1) 
\right] \\= \left\langle\left( \frac 16 \left[ 
\delta_{1,-1}+ \delta_{-1,1}+ \delta_{-1,-1}+ \delta_{1,1}+ 2\delta_{0,0}
\right]\right) ,f\right\rangle
$$
hence the joint distribution is
$$
\frac 16 \left[ 
\delta_{1,-1}+ \delta_{-1,1}+ \delta_{-1,-1}+ \delta_{1,1}+ 2\delta_{0,0}
\right]
$$
or, in more simple notations:
$$
P(\pm 1, \pm 1) = \frac 16\\ P(0,0) = \frac 13
$$
A: 
Random variables $X$  and $Y$  are independent. $X$  takes on values of $−1$  and $1$  with $1/2$   probability each, $Y$  takes on values of $−1,0,1$  with $1/3$   probability each. What is the joint distribution of random variables $Y$  and $XY$ ?
I've calculated the mere distribution of $X\cdot Y$ , but I'm confused with the wording "joint distribution of $Y$  and $XY$ ". What does it mean and what is actually required to do?

You need to find the join probability mass function (and it's support).
You want: $f_{\small Y,XY}(y,z)=\mathsf P(Y = y, XY = z) \\ =\begin{cases} \mathsf P(Y = y)\mathsf P(X = z/y) & : y \in\{-1, 1\}, z=\{-1,1\} \\ \mathsf P(Y=0) & : y=0, z=0 \\ 0 & \text{: elsewhere}\end{cases} $
A: Make a table with four columns labeled $X$, $Y$, $p$, $XY$, and six rows on which you should enter a value for $X$ (hint: $\pm 1$ only), a value for $Y$ (hint: $-1,0,1$ only), the probability that $X$ and $Y$ have the stated values (hint: independence), and the value of $XY$. If you have more than $6$ rows (or fewer!),
check your work very carefully.
Now, erase the $X$ column. You are left with six pairs of values for $(Y,XY)$ and
the corresponding probabilities. Does the same pair occur on two or more rows?
If so, combine them. If not, what you have is the joint pmf of $Y$ and $XY$.
