Let $X,Y$ be two (possibly dependent) random variables with $X,Y>-1$. I would like to evaluate the expected value of a ratio $$\mathbb{E}\left[\frac{1+Y}{1+X}\right]$$ and stuck on some steps; see detail below.
My thought: Using conditonal expectation. For $y>-1$, $$ \begin{align*} \mathbb{E}\left[\frac{1+Y}{1+X}\right] &= \mathbb{E}\left[ \mathbb{E}\left[\frac{1+Y}{1+X} | \ Y=y\right]\right]\\ &= \mathbb{E}\left[ (1+y) \mathbb{E}\left[\frac{1}{1+X} | \ Y=y\right]\right]\\ &\geq \mathbb{E}\left[ (1+y) \frac{1}{\mathbb{E}[1+X| \ Y=y]} \right] \;\;\; \rm{By \ Jensen's \ inequality.} \end{align*} $$ However, I am not sure if the conditional expection (I know if standard expectation, I can, but not sure for conditional one.) can be pulled out? That is, I wonder $$ \mathbb{E}\left[ (1+y) \frac{1}{\mathbb{E}[1+X| \ Y=y]} \right] = ? = \underbrace{\mathbb{E}\left[ 1+y \right]\frac{1}{\mathbb{E}[1+X| \ Y=y]}}_{=(1+y)\frac{1}{\mathbb{E}[1+X| \ Y=y]}}. $$
Any thought/comment is appreciated.
It is not very clear to me the symbol $y$ that you introduce. If the question is if this holds as an equality :
$$ \mathbb{E}\left[ (1+Y) \frac{1}{\mathbb{E}[1+X| \ Y]} \right] = ? = \mathbb{E}\left[ 1+Y \right]\frac{1}{\mathbb{E}[1+X| \ Y]} $$
Than I would argue that it does not make much sense since on the right we have a Y measurable random variable, and on the left a (deterministic) scalar.
This instead would work if $\frac{1}{\mathbb{E}[1+X| \ Y]}$ and $1+Y$ were independent:
$$ \mathbb{E}\left[ (1+Y) \frac{1}{\mathbb{E}[1+X| \ Y]} \right] = ? = \mathbb{E}\left[ 1+Y \right]\mathbb{E} \left[\frac{1}{\mathbb{E}[1+X| \ Y]}\right] $$
but not in a general case.