I have doubts about two related equalities involving conditional expectation:
Let $X,Y,Z$ be random variables; let $Z$ be independent on $X,Y$.
- Is $E[X \mid \beta Z + Y] = E[X \mid Y]$?
- Is $E[\alpha Z \mid \beta Z + Y] = E[\alpha Z \mid \beta Z] = \frac \alpha\beta Z$ ?
Thoughts
Intuitively, both seem correct, as $Z$ is independent on $X,Y$ and does not add information to $X$ (and conversely $Y$ does not add information to $Z$). It is also true that $$E[X \mid Z,Y] = E[X \mid Y]$$
but in this case I am conditioning on the sum of $Z$ and $Y$. How do the conclusions change?