Is it possible to simplify the following expression, using LIE or otherwise?
$\mathbb{E}\left[(1-X)\mathbb{E}(Y|X)+X\mathbb{E}(Z|X)\right]$
where $X, Y \text{and } Z$ are random variables with $X, Y$ and $X, Z$ correlated. At a glance, it looks like it could work down to something like
$\mathbb{E}(1-X)\mathbb{E}(Y)+\mathbb{E}(X)\mathbb{E}(Z)$
or
$\mathbb{E}[(1-X)Y+XZ]$
but neither seems possible. If all these expressions are different, what would be an intuitive way of understanding the difference between them all?
Thanks in advance for the help.
By pushing $1-X$ and $X$ into the inner expectations (check that you understand why this is possible), we have $$E[E[(1-X)Y \mid X] + E[XZ \mid X]] = E[(1-X)Y] + E[XZ] = E[(1-X)Y + XZ].$$
If $E[Y\mid X] = E[Y]$ and $E[Z \mid X] = E[Z]$ (e.g., if $Y$ and $Z$ are each independent of $X$), then you could rewrite your original expression as $E[1-X] E[Y] + E[X] E[Z]$. But this does not hold in general.