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Math Help - E(X(Y-Z))=0 implies E(Y)=E(Z)?

  1. #1
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    E(X(Y-Z))=0 implies E(Y)=E(Z)?

    Hi all, for three random variables, if the expectation of X*(Y-Z) is zero E(X(Y-Z))=0, does it imply the expectation of Y and Z are equal E(Y)=E(Z)? Thanks! A formal proof of your answer is much appreciated.
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  2. #2
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    Re: E(X(Y-Z))=0 implies E(Y)=E(Z)?

    let $E[Y]\neq 0$

    let $E[XY]=0$

    Let $Z=-Y$

    $E[Z]=-E[Y] \neq E[Y]$

    $Y-Z=2Y$

    $E[X(Y-Z)] = E[2XY]=0$

    so

    $E[X(Y-Z)]=0$$~~\large \not \Rightarrow~~$$E[Y]=E[Z]$
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  3. #3
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    Re: E(X(Y-Z))=0 implies E(Y)=E(Z)?

    Dear romsek, thanks so much for your reply.
    What if in addition to E[Y]≠0 (and E[Z]≠0, E[Y]≠0), I have E[XY]≠0 and E[XZ]≠0 in my context? Can E[X(Y−Z)]=0 implies E[Y]=E[Z]? Thanks!
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