Date: Oct 8, 2012 12:38 PM
Author: Maury Barbato
Subject: Independent Random Variables

I am struggling with the following seemingly very simple
probability problem. Let Y, X_1, X_2 three random variables
(taking real values) on the same probability space,
and suppose that Y and X_1 are independent, as well as
Y and X_2. Then, if Y= X_1 + X_2, I think we should
have Y = c a.s. for some constant c in R.
The proof is trivial if X_1 and X_2 are square-integrable,
because in that case Var(Y)=Cov(Y,X_1 + X_2)=0.
But I do not know how to handle the general case.
What do you think about?
Thank you very very much in advance for your help.
My Best Regards,
Maurizio Barbato