My Math Forum Properties of Expected Value, Covariance, and Random Vectors

 October 24th, 2017, 05:21 PM #1 Senior Member   Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0 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.
 October 24th, 2017, 06:11 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,124 Thanks: 1103 what's the meaning of the comma?
 October 24th, 2017, 06:28 PM #3 Senior Member   Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0 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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