Find correlation of x and y, given E(Y|X) and E(X|Y)

Question

Suppose that X and Y are random variables such that E(Y | X) = 7 - (1/4)x and E(X | Y) = 10 - Y . Determine the correlation of X and Y .

Edit:

So far I've got

E(x)=4 E(y)=6

Now I'm trying to find

E(xy) to use in cov(x,y)=E(xy)-E(x)E(y)

V(x)

V(y)

all to use in cor(x,y)=cov(x,y)/(v(x)v(y))^.5

Answer 1

Hint: $E[Y\mid X]$ is the minimum-mean-square-error estimator of $Y$ given the value of $X$. The linear minimum-mean-square-error estimator of $Y$ given the value of $X$ is

$$\hat{Y} = \mu_Y + \frac{\rho\sigma_Y}{\sigma_X}(X-\mu_X).$$

Similar statements apply to $E[X\mid Y]$ etc. Just interchange $X$ and $Y$ in the above formulas.

Now, if $E[Y\mid X]$ is a linear function of $X$ and $E[X\mid Y]$ is a linear function of $Y$, can you use the known forms of the linear minimum-mean-square-error estimators to deduce the value of $\rho$?