Compute $f_X(x)$ and $f_Y(y)$. And are X,Y independent?

Question

Let $(X,Y)$ be a random vector with joint pdf $f_{X,Y}(x, y)$ = $1_{[−1/2,1/2]^2}(x, y)$. Compute $f_X(x)$ and $f_Y(y)$. And are X,Y independent?

I got:

$f_X(x)$ = $\int_{-0.5}^{0.5} dy$ = $1$

$f_Y(y)$ = $\int_{-0.5}^{0.5} dx$ = $1$

Then, we need $f_{X,Y}(x, y)$ = $f_X(x)$$f_Y(y)$ to let $X,Y$ be independent.

Thus, $1 = 1 * 1$, therefore $X,Y$ are independent. It seems to easy so I am not sure if it is correct

Answer 1

The indicator function tell you that the joint distribution is defined in that square.

To get a marginal density you have to integrate the other variable.

$f_X(x)=\int_{-\frac{1}{2}}^{\frac{1}{2}} dy=1$

Now you have not finished yet...you have to indicate X domain...

$f_X(x)=\mathbb{1}_{[-\frac{1}{2};\frac{1}{2}]}(x)$

Do the esame for the other rv and conclude