exponenial distribution
Solution 1:
The conditional distribution of $X$ given $Y$ is uniform on $[Y, 3Y]$, so $\mathbb{E}[X\vert Y] = 2Y$. Then $$\mathbb{E}[X] =\mathbb{E}[\mathbb{E}[X\vert Y]] = 2\mathbb{E}[Y] $$ and $$\mathbb{E}[XY] = \mathbb{E}[\mathbb{E}[X\vert Y]Y] = 2\mathbb{E}[Y^2].$$ Finally $$\text{Cov}(X, Y) = 2\mathbb{E}[Y^2] - 2\mathbb{E}[Y]^2 = 2\text{Var}(Y) = \frac{2}{9}. $$