It's me again, sorry... I've a Problem with a conditional expectation: are two random variables and is just a constant. Does the follwoing hold? ? Or does it just hold if and are independent? Thanks in advance!
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Hello, In this specific case, E[X1X2|X1=c]=E[cX2|X1=c]=cE[X2|X1=c] And in general, X and Y not necessarily independent, E[XY|X]=XE[Y|X]
Thank you so much. I'm a litte bit counfused right now.. It's just a special case of the well known property of the conditional expectation: if is measurable and is a field Right?
Exactly ! When one conditions with respect to a random variable : E[.|X], it's actually conditionned with respect to the sigma-field/algebra generated by the random variable : And by definition, X is -measurable
If you also have independence, then
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