Proof: E[X|Y]=0 implies COV[X,Y]=0
I was thinking maybe the law of total covariance or tower rule but couldn't come up with the proof
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$\begingroup$ Hint: consider the expectation of $E[X\mid Y]Y.$ $\endgroup$– whuber ♦Commented Jan 25, 2021 at 17:36
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$\begingroup$ It's a direct application of tower rule. $\endgroup$– StubbornAtomCommented Jan 25, 2021 at 20:45
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$\begingroup$ Start by writing out the covariance definition out $\endgroup$– Three DiagCommented Jan 25, 2021 at 20:49
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$\begingroup$ If you would like details, see the equation immediately preceding the "Conclusions" section in my post at stats.stackexchange.com/questions/71260/… and let $\rho=0.$ $\endgroup$– whuber ♦Commented Jan 26, 2021 at 14:02
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