Checking if jointly CRVs are independent via the marginal density

Question

I'm trying to determine whether the following continuous random variables are independent or not given their joint probability density.

$f_{X,Y}(x,y) = 2(x+y)$ for $ 0\le y\le x\le 1$

I calculated the marginal densities as follows:

$f_x(x) = \int_0^1{2(x+y)} dy$ = $ 2x+1$

Using the same method for $f_y(y)$ I got $f_y(y) = 2y+1$

Since $f_{X,Y}(x,y)\neq f_X(x)f_Y(y) $ they are not independent.

My concern is that my calculation of the marginal densities is incorrect, could someone verify whether my method is correct? Or if this is the best way to check if the two are independent.

Answer 1

My concern is that my calculation of the marginal densities is incorrect, could someone verify whether my method is correct? Or if this is the best way to check if the two are independent.

It is not correct. $$f_{\small Y}(y)=\int_y^1 2(x+y)\,\mathbf 1_{0\leqslant y\leqslant 1}\,\mathrm d x\\f_{\small X}(x)=\int_0^x 2(x+y)\,\mathbf 1_{0\leqslant x\leqslant 1}\,\mathrm d y$$

However, there is a easier way.   The support for the joint p.d.f. is $0\leqslant y\leqslant x\leqslant 1$.   That is all you need to know.

Without even integrating, you can assert that: $$\begin{align}\mathsf P(Y>1/2,X<1/2)&=0\\\mathsf P(Y>1/2)&>0\\\mathsf P(X<1/2)&>0\end{align}$$