Statistics - correlation coefficient

Question

I have a statistics question and have calculated the part (a) answer. But I have no idea on part (b) as i have the no clue on $E(XY)$ to calculate the cov($X$.$Y$). I write to seek for advice on part (b). Please find below for the details and my calculation on part (a).

Let X and Y be random variables with $E(X|Y)=2Y$, Var($X$|$Y$)=$6Y$^$2$, $E(Y)=7$, Var($Y$)$=10$.

(a) Find the mean and variance of X.

(b) Find the correlation coefficient between X and Y.

Ans for (a)
E(X)
=E(E(X|Y))
=E(2Y)
=2E(Y)
=14

Var(X)
= E(Var(X|Y))+Var(E(X|Y))
= E(6Y²)+Var(2Y)
= 6E(Y²)+4Var(Y)
= 6(Var(Y)+(E(Y))²)+4Var(Y)
= 6(10+7²)+4x10
= 394

Please help to advise the part (b) solution. Thanks!

Answer 1

Hint$$\mathbb{E}[XY]=\mathbb{E}[\mathbb{E}[XY|Y]]=\mathbb{E}[Y\,\mathbb{E}[X|Y]]=2\mathbb{E}[Y^2]=2(10+7^2)=118$$ $$\text{Cov}(X,Y)=\mathbb{E}[XY]-\mathbb{E}[X]\mathbb{E}[Y]=118-98=20$$