Let $(X,Y)$ be a random vector whose density $f$ is given as follows: $$f(x,y) = \frac{\sqrt{3}}{2\pi}\exp\left(\frac{-1}{2}(4x^2+2xy+y^2)\right)$$ $\forall (x,y)\in \mathbb{R}^2.$
With this information, I determined the density of $X$ and $Y.$ $$f_X(x) = \frac{\sqrt{3}}{\sqrt{2\pi}}\exp\left(-\frac{3x^2}{2}\right)$$ and $$f_Y(y) = \frac{\sqrt{3}}{2\sqrt{2\pi}}\exp\left(-\frac{3y^2}{8}\right).$$ I see that $f(0,0)\neq f_X(0)f_Y(0)$, but I am not sure whether this is enough to conclude that $X$ and $Y$ are independent random variables.