Independence, conditional expectations and uncorrelatedness

Problem

We know that if \(X\) and \(Y\) are two independent random variables then \(E\left( Y \mid X \right)=E\left( Y \right)\) . Let’s investigate if the opposite is true. Say that there are three possibilities,

• \(X=0,Y=-1\) (probability ¼)
• \(X=0,Y=1\) (probability ¼)
• \(X=1,Y=0\) (probability ½)

1. Explain why \(X\) and \(Y\) are not independent.
2. Find \(E\left( Y \right)\)
3. Find \(E\left( Y \mid X=0 \right)\)
4. Find \(E\left( Y \mid X=1 \right)\)

We see that \(E\left( Y \mid X \right)=E\left( Y \right)\) even though \(X,Y\) are dependent. This demonstrates that the opposite is not true .

However, you can show that if \(E\left( Y \mid X \right)=E\left( Y \right)\) then \(X\) and \(Y\) must be uncorrelated (which is a weaker condition than independent). The opposite to this result is also false. Even if \(X\) and \(Y\) are uncorrelated it does not follow that \(E\left( Y \mid X \right)=E\left( Y \right)\) . \(E\left( Y \mid X \right)=E\left( Y \right)\) is “in between” the stronger independence and the weaker uncorrelated:

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

None of the arrows go in the other direction.