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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?

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  • $\begingroup$ XY can be calculated as the product of X and Y since X and Y are independent. Then you can consider the joint PDF of Y and XY as the product of Y and XY times their covariance. $\endgroup$ – Tony S.F. Nov 3 '14 at 22:47
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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: $$ P(\pm 1, \pm 1) = \frac 16\\ P(0,0) = \frac 13 $$

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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 $XY$.

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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} $

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