Joint probability mass function - forming a table

$X_1$ and $X_2$ are independent random variables with distribution given by $P(X_i = −1) = P(X_i = 1) = 1/2$ for $i = 1,2$

Find the joint probability mass function of $X_1$ and $X_2$

I think the entire table would have probabilities equal to $1/4.$ I thought that since they are independent, we just need to multiply $(0.5)(0.5)$

Turns out, I was wrong. For $P(X_1 = - 1, P(X_2 = 1),$ the value is $1/2.$ How?

The second part says that $Y= X_1 X_2$ and find the joint pmf of $X_1$ and $Y.$ I'm completely lost here because how do I fill in the table. What will be the probabilities?

Solution 1:

Hint: For your second question, $$ P\left(X_1=x, Y=y\right)=P\left(X_1=x, X_2=\frac{y}{x_1}\right)\ , $$ so you can read the joint pmf of $\ X_1\ $ and $\ Y\ $ straight off the table you construct for the joint pmf of $\ X_1\ $ and $\ X_2\ $.