Conditional expectations

Problem

\(X,Y\) are two random variables. There are only two possibilities, \(X=0,Y=0\) or \(X=1,Y=1\) and these are equally likely.

  1. Are \(X\) and \(Y\) independent random variables?
  2. Find \(E(Y)\)
  3. Find \(E\left( Y \mid X=0 \right)\)
  4. Find \(E\left( Y \mid X=1 \right)\)
  5. Explain why \(E\left( Y \mid X=2 \right)\) would make no sense.

Solution

  1. No, there is a dependence
  2. \(0×0.5+1×0.5=0.5\)
  3. If \(X=0\) , \(Y\) must be zero so \(E\left( Y \mid X=0 \right)=0\)
  4. If \(X=1\) , \(Y\) must be one so \(E\left( Y \mid X=1 \right)=1\)
  5. We cannot condition on something that cannot happen. \(X=2\) is not possible.