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$X \sim U(0,2)$

$Y \sim U(0,2)$

If you could clarifiy the last transition...

$Var(XY) = E[(XY)^2] - E[XY]^2 = E[X^2Y^2] - E[X]^2E[Y]^2 = E[X^2]^2 - 1 = (V(X)+E[X]^2)^2 - 1$

How do you get from $E[X^2]^2$ to $(V(X)+E[X]^2)^2$

Thanks.

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1 Answer 1

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Well as you can easily see from expanding $\mbox{Var}(X) = \mathbb{E}\left[ \left(X - \mathbb{E}[X]\right)^2\right]$ we have the identity \begin{align*} \mbox{Var}(X) = \mathbb{E}\left[ X^2 \right] - \mathbb{E}\left[X \right]^2, \end{align*} and so \begin{align*} \mathbb{E}[X^2] = \mbox{Var}(X) + \mathbb{E}[X]^2 \end{align*} which is the expression inside the parenthesis.

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