Consider two random variables \(X,Y\in L^2(\mathbb {P})\). Prove that if \(\mathbb {E}[X\mid Y]=Y\) and \(\mathbb {E}[X^2\mid Y]=Y^2\), then \(X=Y\) a.s.
Question
Math 6010 • Linear Models
2004 • Question 9
Question
2004 • Question 9
Consider two random variables \(X,Y\in L^2(\mathbb {P})\). Prove that if \(\mathbb {E}[X\mid Y]=Y\) and \(\mathbb {E}[X^2\mid Y]=Y^2\), then \(X=Y\) a.s.