Question

Prove that if $E[Y\mid X=x]=E\left[Y\right]$for all $x$, then $X$and $Y$are uncorrelated; give a counterexample to show that the converse is not true.

Hint: Prove and use the fact that $E\left[XY\right]=E\left[XE\right[Y\mid X\left]\right]$.

Short answer

We prove that,$E[Y\mid X=x]=E\left[Y\right]$for all$x$

Step-by-step solution

Given information

Given in the question that, we have to prove that$E[Y\mid X=x]=E\left[Y\right]$for all$x$

Explanation

Let's find the covariance between $X$and $Y$. We have that

$\mathrm{Cov}(X,Y)=E\left(XY\right)-E\left(X\right)E\left(Y\right)$

Since we have that $E(Y\mid X=x)=E\left(Y\right)$for all $x$es, we have that

$E(Y\mid X)=E\left(Y\right)$

Which implies

Hence

$\mathrm{Cov}(X,Y)=0$

Disapprove the converse

Now, let's disapprove the converse. Consider random variable $X~\mathcal{N}(0,1)$and define $Y=X^2$. We have that

$\mathrm{Cov}(X,Y)=\mathrm{Cov}\left(X,X^2\right)=E\left(X^3\right)-E\left(X\right)E\left(X^2\right)$

Observe that $X^3$is symmetric around zero, so $E\left(X^3\right)=0$. Finally, we see that $Cov(X,Y)=0$. But, we have that

$E(Y\mid X)=E\left(X^2\mid X\right)=X^2$

which is not obviously equal to constant $E\left(Y\right)=E\left(X^2\right)$.

Final answer

We prove that, $E[Y\mid X=x]=E\left[Y\right]$for all $x$

And from the above example we prove that$E\left(Y\right)=E\left(X^2\right)$