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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let $E[Y]\neq 0$
$E[Z]=-E[Y] \neq E[Y]$
$E[X(Y-Z)] = E[2XY]=0$
$E[X(Y-Z)]=0$$~~\large \not \Rightarrow~~$$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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