How to check if two random variables are independent without calculating their density functions?

Let $$D = \{(x,y): x^2 + y^2 \leq 1 \}$$ and X,Y two random independent variables with joint density function: $$f(x,y) = c e^{-(x^2+y^2)}\mathbb{1}_D(x,y)$$ I want to understand if X and Y are two independent random variables, without calculating $$f_X(x)$$ and $$f_Y(y)$$. However, I just know that method, any tips on how to approach this?

• Intuitively: Knowing about $X$ tells you about $Y$ in this example, since $X,Y$ lie in the unit circle: $Y \leq \sqrt{1 - X^2}$. So they are not independent. – Jair Taylor Oct 28 '18 at 21:18
• @JairTaylor Do we know that both $X$ and $Y$ lie within the disk? Why can't I have $X=\frac{1}{2}$ and $Y=10$? – John Douma Oct 28 '18 at 21:33
• @JohnDouma No, the density function is $0$ outside the disk, so $X^2 + Y^2 \leq 1$ almost surely. – Jair Taylor Oct 28 '18 at 22:12

Hint: If $$X$$ and $$Y$$ are independent and $$\mathbb P(a \le X \le b) > 0$$ and $$\mathbb P(c \le Y \le d) > 0$$, then $$\mathbb P(a \le X \le b,\; c \le Y \le d) > 0$$.