Expected value of conditional probability

probability

I'm working on a proof.

If X and Y are independent, then

$E(Y|X) = E(Y)$

X, Y are discrete random variables. And we assume that their joint marginal probability mass functions and expectations exists.

I'm pretty lost. My initial idea is to use the following definition:

$E(g(Y_1) | Y_2=y_2) = \sum_{all\;y_1} g(y_1)p(y_1|y_2)$

Should I use that, and if I should, how to proceed?

Hope you can help!


the proof is quite immediate

$$E(Y|X)=\Sigma_y y\cdot p(y|x)=\Sigma_y y \cdot p(y)=E(Y)$$

this because by independence $p(y|x)=p(y)$

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