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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, YZ)} + Cov{E(XZ), E(YZ)} = Cov(X, E(YX)) 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: 1,944 Thanks: 1011 
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(YX)) 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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covariance, expected, properties, random, vectors 
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