Consider an experiment in which three fair dice are tossed simultaneously and independently. Let $Z_1,Z_2,Z_3$ be the values resulting from the three tosses. Define $X=Z_{21}+Z_{22}−Z_{33}$ and $Y=4X−7$. What is the correlation coefficient between $X$ and $Y$?
Ok so really my problem is how to calculate covariance in general. Once I find the covariance, I know how to calculate the correlation coefficient. I know that $\mathrm {Cov}(X,Y)=E(XY)-E(X)E(Y)$, and calculated $E(X)= -18.375$ and $E(Y)= -80.5$ but I don't know how to calculate $E(XY)$.
Any help?
HINT: $Cov(X,Y)=Cov(X,4X-7)=4Cov(X,X)-Cov(X,7)=4 ~~ Var(X)$
$Corr(X,Y)=\dfrac{4~Var(X)}{\sqrt {Var(X)~.Var(4X-7)}}=1$
Note that it is also justified since $X$ and $Y$ have the perfect linear relationship $Y=4X-7$