S13.1 Conditional Expectation Properties  Summary and Q&A
TL;DR
Conditional expectations have intuitive properties, such as being able to treat certain variables as constants and the equality of conditional expectations when the random variables have invertible functions.
Questions & Answers
Q: What is the intuitive meaning of conditioning on a variable?
Conditioning on a variable means that its value is known, eliminating randomness, and allowing for the treatment of certain functions as constants.
Q: How does one establish the result of pulling constants outside the expectation formally?
In the discrete case, one shows that two random variables are equal by considering specific outcomes and demonstrating that the random variables take the same value regardless of the value of the conditioned variable.
Q: Can the result of pulling constants outside the expectation be applied to continuous or general random variables?
The result still holds for continuous or general random variables, but a rigorous proof is more complex and involves technical details beyond the scope of this content.
Q: What are the properties of conditional expectations when the random variables have invertible functions?
When the random variables have invertible functions, the conditional expectations will be the same, as the conditional distribution of X should be the same regardless of which variable is conditioned on.
Summary & Key Takeaways

Conditional expectations allow for the treatment of certain variables as constants and can be pulled outside the expectation.

When the random variables have invertible functions, the conditional expectations will be equal.

The details of formal proof for continuous or general random variables are complex but the intuitive properties still hold.