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If $E\left(Y|X\right) = E(Y)$ can we state that $X$ and $Y$ are independent? And vice verse, if $X$ and $Y$ are independent can we state that $E\left(Y|X\right) = E(Y)$?

falkor86
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    See e.g. https://stats.stackexchange.com/questions/188242/for-intuition-what-are-some-real-life-examples-of-uncorrelated-but-dependent-ra/188248#188248 for a related (counter-)example, in that it addresses uncorrelated but dependent random variables. – Christoph Hanck May 27 '21 at 09:07

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If conditional expectation is equal to unconditional expectation does that mean the random variables are independent?

Not necessarily.

If $X$ and $Y$ are independent we can say that surely $E[Y|X]=E[Y]$ and $E[X|Y]=E[X]$. If $E[Y|X]=E[Y]$ we can say that $Y$ is mean independent of $X$ but not necessarily them are completely independent; Indeed it is not implied neither $E[X|Y]=E[X]$.

markowitz
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  • Thanks, I found this too https://math.stackexchange.com/questions/2815227/is-there-any-counterexample-to-show-that-x-y-are-two-random-variables-and-ex?rq=1 for mean independent random variables that are not independent. – falkor86 May 27 '21 at 09:45