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October 24th, 2017, 06:21 PM   #1
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Exclamation Properties of Expected Value, Covariance, and Random Vectors

Given a random vector (X, Y), prove the following:

E{Cov(X, Y|Z)} + Cov{E(X|Z), E(Y|Z)} = Cov(X, E(Y|X))

when Z = X.

After substituting Z with X, I'm struggling to simplify the left hand side of the expression. There must be some properties of expected value that I don't know about, but I'm not seeming to come any closer when I research. There must be something simple that I'm missing. Will somebody please explain the simplification? Thank you.
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October 24th, 2017, 07:11 PM   #2
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what's the meaning of the comma?
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October 24th, 2017, 07:28 PM   #3
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The comma is to separate the random variables which are the arguments of the covariance. So Cov(X, E(Y|X)) is the covariance of the random variable X with the expected value of the random variable Y given X.

Here's the whole problem:

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