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Thread: conditional expectation

  1. #1
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    conditional expectation

    It's me again, sorry...

    I've a Problem with a conditional expectation:

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

    Does the follwoing hold?

    $\displaystyle E[X_1X_2|X_1=c]=cE[X_2|X_1=c]$?

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

    Thanks in advance!
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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:
    $\displaystyle E[XY|\mathcal{G}]=XE[Y|\mathcal{G}]$ if $\displaystyle X$ is $\displaystyle \mathcal{G}-$measurable and $\displaystyle \mathcal{G}$ is a $\displaystyle \sigma-$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 :

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

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

    If you also have independence, then

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