I am doing an assignment and arrived at a question that I could not figure out and was hoping for some hints. Let X and Y be two random variables with common variance $a^2$ (a > 0). Suppose that $E ( Y | X ) = X + 1$. Determine the correlation coefficient.
I know that $E(Y) = E(X)+1$, but no matter what I do I cannot find a way to calculate the covariance as I do not know E(XY) .
$Cov(X,Y)=E(XY)-E(X)E(Y)$.
Now $E(XY)=E(E(XY|X))=E(XE(Y|X))=E(X(X+1))=E(X^2)+E(X)=Var(X)+(E(X))^2+E(X)=a^2+(E(X))^2+E(X)$
$E(X)E(Y)=E(X)E(E(Y|X))=E(X)E(X+1)=[E(X)]^2+E(X)$
Therefore, $Cov(X,Y)=a^2$
Thus, correlation coefficient = $\dfrac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}=\dfrac{a^2}{a^2}=1$
You could also have noticed that you are in the linear regression set-up where $Y=1+X+\epsilon$ and coefficient of $X$ is $1$, so as $Var(X)=Var(Y)$, you have $b_{yx}=\rho\dfrac{s_y}{s_x}=\rho$ but $b_{yx}=1$ hence $\rho=1$.