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Math Help - conditional expectation

  1. #1
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    conditional expectation

    It's me again, sorry...

    I've a Problem with a conditional expectation:

    X_1,X_2 are two random variables and c>0 is just a constant.

    Does the follwoing hold?

    E[X_1X_2|X_1=c]=cE[X_2|X_1=c]?

    Or does it just hold if X_1 and X_2 are independent?

    Thanks in advance!
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  2. #2
    Moo
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    Re: conditional expectation

    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]
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    Re: conditional expectation

    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:
    E[XY|\mathcal{G}]=XE[Y|\mathcal{G}] if X is \mathcal{G}-measurable and \mathcal{G} is a \sigma-field

    Right?
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    Re: conditional expectation

    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 :

    E[.|\sigma(X)]:=E[.|X]

    And by definition, X is \sigma(X)-measurable
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  5. #5
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    Re: conditional expectation

    If you also have independence, then

    \mathbb E(XY|X)=X\mathbb E(Y|X)=X\mathbb E(Y)
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