Discussion Overview
The discussion revolves around the properties of conditional expectations in probability theory. Participants explore whether certain conditions regarding expected values of a random variable hold true under specific circumstances, particularly focusing on the implications of conditional expectations given different subsets of information.
Discussion Character
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant proposes that if E(v|x1) = b and E(v|x2) = b, it follows that E(v|x1,x2) = b, seeking clarification on this implication.
- Another participant argues against the converse, stating that E(v|x1,x2) = b does not necessarily imply E(v|x1) = b and E(v|x2) = b, suggesting that counterexamples exist.
- A participant suggests verifying the original claim by breaking down the definitions of conditional expectations and using algebraic manipulation.
- One participant provides a specific example involving rolling two dice, where v represents the total and x1 and x2 represent specific outcomes of the dice.
- Subsequent posts question whether the provided example serves as a valid counterexample to the original claim.
Areas of Agreement / Disagreement
Participants express disagreement regarding the validity of the converse statement about conditional expectations. The original claim about the implications of E(v|x1) and E(v|x2) remains a point of contention, with no consensus reached.
Contextual Notes
Participants reference the definitions of conditional expectations and the need for specific examples to illustrate their points, indicating that further mathematical exploration may be necessary to fully resolve the questions posed.