Question about Law of Iterated Expectations

Question

So I am trying to show that the following holds:$$\mathbb{E}[Y \times \mathbb{E}(X|Z)]=\mathbb{E}[X \times \mathbb{E}(Y|Z)]$$ If my intuition is correct, I believe that this can be solved using the Law of Iterated expectations. (I am assuming both expectations $<+\infty$) When trying to solve, I started with the LHS. I was trying to make use of $\mathbb{E}[\mathbb{E}(X|Z)=\mathbb{E}(X)$, but I don't think I can since it is being multiplied by $Y$.

I would appreciated any hints.

Answer 1

Working with each term

1. $\mathbb{E}[Y \mathbb{E}(X|Z)]=\mathbb{E}(\ \mathbb{E}[\ Y \mathbb{E}(X|Z)\ |\ Z]\ )=\mathbb{E}(\mathbb{E}(X|Z)\mathbb{E}[Y|Z])$
2. $\mathbb{E}[X \mathbb{E}(Y|Z)]=\mathbb{E}(\ \mathbb{E}[\ X \mathbb{E}(Y|Z)\ |\ Z]\ )=\mathbb{E}(\mathbb{E}(Y|Z)\mathbb{E}[X|Z])$

you have equality.