Joint probability density function and independence The joint probability density function of the random variables $X_1$ and $X_2$ is given by$f_{x_1 x_2}(x_1,x_2)= \left\{\begin{matrix}
(1/6) \cdot x_1x_2,1\leq x_1 \leq 3 \quad and \quad 1\leq x_2\leq 2\\ 
0, otherwise\end{matrix}\right.$

And let the variable Y be defined as $ Y = X_1 X_2^2$

How do I find out if $X$ and $Y$ are independent and if they are uncorrelated? And how do i find he mean of $E[Y]$ ?
Hint 1: If $X_1$ and $X_2$ are independent then $\mathbb E(X_1X_2^2)=\mathbb E(X_1)\cdot \mathbb E(X_2^2)$

Hint 2: $f_{X_2}(x_2)=\int_1^3 \frac16\cdot x_1\cdot x_2 \, \ dx_1$ 

Hint 3: $\mathbb E(X_2^2)= \int_1^2 x_2^2 \cdot f_{X_2}(x_2)\, \ dx_2$
Hint 1: If $X_1$ and $X_2$ are independent then $\mathbb E(X_1X_2^2)=\mathbb E(X_1)\cdot \mathbb E(X_2^2)$

Hint 2: $f_{X_2}(x_2)=\int_1^3 \frac16\cdot x_1\cdot x_2 \, \ dx_1$ 

Hint 3: $\mathbb E(X_2^2)= \int_1^2 x_2^2 \cdot f_{X_2}(x_2)\, \ dx_2$