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?
- If \(X,Y\) are independent then \(E\left( Y \mid X \right)=0\)
- If \(E\left( Y \mid X \right)=0\) then \(X,Y\) are uncorrelated
- If \(X,Y\) are independent then \(X,Y\) uncorrelated
- If \(X,Y\) are uncorrelated then \(X,Y\) independent
- If \(X,Y\) are correlated then \(E\left( Y \mid X \right)=0\) cannot be true
Solution
Remember
\[X,Y \textrm{ are independent } \implies E\left( Y \mid X \right)=E\left( Y \right) \implies X,Y \textrm{ are uncorrelated }\]
- True, first arrow
- True, second arrow
- True
- False, cannot go left
- True. If \(E\left( Y \mid X \right)=0\) was true then \(X,Y\) would have to be uncorrelated.