# a question about conditional expectation

Suppose that X and Y are random variables such that E(Y/X)=aX+b,how determine expressions for a and b in terms of E(X),E(Y),Var(X),and Cov(X,Y).assuming that Cov(X,Y) exists and Var(X)>0.

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Using the tower property, recall that $E[E[Y|X]] = E[Y]$. You should try to work that out in order to obtain an equation in terms of $E[X]$ and $E[Y]$,

$$E[Y] = E[E[Y|X]] = E[aX + b] = aE[X] + b$$

Also, recall that $Cov(X,Y) = E[XY] - E[X]E[Y]$. Hence you can try to write $E[XY]$ in terms of $Var[X]$ and $E[X]$ to get another equation. Recall that $E[XY] =E[E[XY|X]] = E[X \cdot E[Y|X]]$,

$$E[XY] = E[E[XY|X]] = E[X \cdot E[Y|X]] = E[aX^{2} + bX] = a(Var[X]+E[X]^2) + bE[X]$$

Using this expression, obtain:

$$Cov(X,Y) = a(Var[X]+E[X]^2) + bE[X] - E[X]E[Y]$$

Finally, you can obtain the desired relation solving for $a$ and $b$ using the two equations.

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oh,I get it,thanks! –  perry zhu Jul 1 '12 at 6:15