Problem: Conditional expectation, uncorrelatedness and independence

Problem

\(X,Y\) are two random variables.

• Show that \(E\left( X \right)=0\) if \(X,Y\) are independent random variables and \(E\left( Y \right)=0\) .
• If \(E\left( X \right)=0\) , show that the unconditional expected value must be zero as well, \(E\left( Y \right)=0\) .
• If \(E\left( X \right)=0\) , show that \(X,Y\) are uncorrelated . Hint: \(Cov\left( X,Y \right)=E\left( XY \right)-E\left( X \right)E(Y)\) . Find \(E(XY)\) using the law of iterated expectations.
• Suppose that \(X,Y\) are uncorrelated. Will this imply that \(E\left( X \right)=0\) ?

Hint: Consider \(f(x,y)\) defined by the following table:

• Suppose that \(E\left( X \right)=0\) . Does this imply that if \(X,Y\) are independent random variables?
• Hint: Consider \(f(x,y)\) defined by the following table:

Taken together:

\(X,Y\) independent + \(E\left( Y \right)=0\) \(⟹E\left( X \right)=0 ⟹\) \(X,Y\) uncorrelated

but no arrow goes in the opposite direction. \(E\left( X \right)=0\) is an assumption “ in between ” independence (which is often too strong) and uncorrelatedness (which is often not enough)