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)
- Explain why \(X\) and \(Y\) are not independent.
- Find \(E\left( Y \right)\)
- Find \(E\left( Y \mid X=0 \right)\)
- 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.
Solution
- \(Y\) depends on \(X\) (and vice versa). For example, if \(X=1\) then we know that \(Y\) must be zero.
- \(\left( -1 \right)×0.25+1×0.25+0×0.5=0\)
- If \(X=0\) then \(Y=-1 \) or \(Y=1\) and these are equally likely (50%) so \(E\left( Y \mid X=0 \right)=\left( -1 \right)×0.5+1×0.5=0\)
- If \(X=1\) then \(Y=0\) for sure so \(E\left( Y \mid X=1 \right)=0\)
Since \(E\left( Y \mid X=0 \right)=0\) and \(E\left( Y \mid X=1 \right)=0\) and \(X\) can only take values 0 and 1 we have \(E\left( Y \mid X \right)=0\) .