I feel like this function is not right but I can not prove whether it is right or wrong. Could anyone give a hand explaining why it is right or why it is wrong?
2026-04-25 17:37:22.1777138642
How can I prove that E[X + Y |X,Y ] = E[X|Y ] + E[Y |X]?
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Your identity is false (e.g. consider $X,Y$ independent). Whatever is conditioned on is fixed so
$$E[X+Y|X,Y]=X+Y.$$