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

1. ## 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.

2. ## 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]$