I want to compare (if possible) the following two terms: (1) $\mathrm{Var}(\mathrm{E}[Y | X_1])$ and (2) $\mathrm{Var}(\mathrm{E}[Y | X_1, X_2])$. If they are not comparable as they are, the following assumptions can also be made (each is a seperate case):
- $X_1$ and $X_2$ are independent.
- $X_1$ and $X_2$ are jointly Gaussian and each has standard distribution with $\mathcal{N}(0, 1)$.
I think both (1) and (2) are constants and there should be a way to compare them. I do not know where to start. Any leads to a solution will be appreciated. Thanks in advance!