My Goal
I am trying to figure out what $\Bbb E[X|Y]\Bbb E[Y]$ simplifies to.
My Work So Far
I have the following train of thought
$$\Bbb E[X|Y] = \sum_i x_i \Bbb P(X=x_i|Y=y_j)$$ $$\Bbb E[Y] = \sum_j y_j \Bbb P(Y=y_j)$$
Taking these and multiplying
\begin{align} \Bbb E[X|Y] \Bbb E[Y] & = \left(\sum_i x_i \Bbb P(X=x_i|Y=y_j)\right)\left( \sum_j y_j \Bbb P(Y=y_j)\right)\\ &= \sum_j \sum_i x_i y_j \Bbb P(X=x_i|Y=y_j)\Bbb P(Y=y_j)\\ &= \sum_j \sum_i x_i y_j \Bbb P(X=x_i) \\ &= \sum_j y_j \Bbb E[X] \end{align}
But this isn't very elegant, so I feel like this can't be right.
My Question
Am I on the right track? If so, what are the steps I am missing? If not, why and how should I approach this?
Edit:
I think I may have made a simple mistake.
I think it should probably be
\begin{align} \Bbb E[X|Y] \Bbb E[Y] & = \left(\sum_i x_i \Bbb P(X=x_i|Y=y_j)\right)\left( \sum_j y_j \Bbb P(Y=y_j)\right)\\ &= \sum_j \sum_i x_i y_j \Bbb P(X=x_i|Y=y_j)\Bbb P(Y=y_j)\\ &= \sum_i x_i \Bbb P(X=x_i) \\ &= \Bbb E[X] \end{align}