1. ## Do these texts contradict eachother?

Do these two texts contradict each other?

Or does it mean that not all cases where E[Y | X] = E[Y] are independent, but if X and Y are independent, then E[Y | X] = E[Y] ?

2. Originally Posted by billym
Do these two texts contradict each other?
No, they do not

Or does it mean that not all cases where E[Y | X] = E[Y] are independent,
Unless we know that X and Y are independent, E[Y | X] = E[Y] doesn't prove they are independent. E[Y | X] = E[Y] could be just a coincidence.

but if X and Y are independent, then E[Y | X] = E[Y] ?
Yes, if X and Y are independent, X doesn't effect Y, and vice versa. Thefore E[Y | X] = E[Y]

3. Originally Posted by billym

Or does it mean that not all cases where E[Y | X] = E[Y] are independent, but if X and Y are independent, then E[Y | X] = E[Y] ?
Yes, it means what you've just written.
It there would be this equivalence :

for any [some conditions over] f, E[f(Y)|X]=E[Y] <=> X and Y are independent.

So they do not contradict each other