Suppose $X$ and $Y$ are random variables. As I understood it, mean independence is defined as follows
$Y$ is mean-independent of $X$ iff $\Bbb E[Y|X] = \Bbb E[Y]$
But my professor in class gave the following definition in class yesterday
$Y$ is mean-independent of $X$ iff $\Bbb E[Y|X] = c$, where $c \in \Bbb R$
Is this the same thing?
Like suppose $\Bbb E[Y|X] = c$, where $c \ne \Bbb E[Y]$.
EDIT:
Actually, my last statement can't be true by the law of iterated expectations since
$$E_X[E[Y|X]] = E[Y]$$
And by that law, then if $E[Y|X] = c$, $E_X[E[Y|X]] = E[Y] = c$