On the conditional expectation E(X | X + W) in the case of independent random variables X, W

✍ Scribed by Ehrhard Behrends

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 217 KB
• Volume: 41
• Category: Article
• ISSN: 0167-7152
• DOI: 10.1016/s0167-7152(98)00192-8

✦ Synopsis

Let X, W be independent real-valued random variables with finite expectation and E(W)=-0. We prove that only in the case W = 0 the conditional expectation E(X IX + W) coincides with X + W. The result is a consequence of the following cancellation theorem: Let P,Q,R be Borel probability measures on the real line such that the support of Q resp. R is contained in {x ~<0} resp. {x I>0}; then P, Q = P, R implies that Q = R(=60). (~