I should find the real constants $a$ and $b$ such as to minimize this in the case in which $\mathbb{E}[X] = \mathbb{E}[Y] = 0$:
$$ \begin{align} \mathbb{E}[(Y - (a + b X))^2 ] &= \mathbb{E}[Y^2] + \mathbb{E}[(a + b X)^2] - \mathbb{E}[2 Y (a + b X)] = \\ &= \mathbb{E}[Y^2] + a^2 + b^2 \mathbb{E}[X^2] + 2ab \mathbb{E}[X] - 2a \mathbb{E}[Y] - 2b \mathbb{E}[XY] = \\ &= \mathbb{E}[Y^2] + a^2 + b^2 \mathbb{E}[X^2] - 2b \mathbb{E}[XY] \end{align} $$
I'm trying to understand why I should choose $a = 0$ (this is the choice in my handout).