Independence, conditional expectations and uncorrelatedness

Problem

\(X\) and \(Y\) are two random variables with \(E\left( X \right)=E\left( Y \right)=0\) . What is true?

  1. If \(X,Y\) are independent then \(E\left( Y \mid X \right)=0\)
  2. If \(E\left( Y \mid X \right)=0\) then \(X,Y\) are uncorrelated
  3. If \(X,Y\) are independent then \(X,Y\) uncorrelated
  4. If \(X,Y\) are uncorrelated then \(X,Y\) independent
  5. If \(X,Y\) are correlated then \(E\left( Y \mid X \right)=0\) cannot be true

Solution

Remember

\[X,Y \textrm{ are independent } \implies E\left( Y \mid X \right)=E\left( Y \right) \implies X,Y \textrm{ are uncorrelated }\]

  1. True, first arrow
  2. True, second arrow
  3. True
  4. False, cannot go left
  5. True. If \(E\left( Y \mid X \right)=0\) was true then \(X,Y\) would have to be uncorrelated.