$A$ has distribution $U(-1,1)$. $Y = AX$. $X$ and $A$ are independent (and can assume $X^2$ and $A^2$ are independent). Compute $E[(Y-E[Y|X])^2].

Question

Please could someone check my answer for the following question: Let $X$ be a random variable with finite variance. $A$ has distribution $U(-1,1)$. $Y = AX$. $X$ and $A$ are independent (and can assume $X^2$ and $A^2$ are independent). Compute $E[(Y-E[Y|X])^2].

Here's what I've done: $E[Y|X] = E[AX|X] = XE[A|X] = E[A]X$. $E[(Y-E[Y|X])^2] = E[(AX-E[A]X)^2] = E[A^2X^2+(E[A])^2X^2-2AX^2E[A]] = E[A^2]E[X^2] - (E[A])^2E[X^2] = \frac{2}{9}E[X^2]$