For a linear model $y_i=\beta_0+\beta_1x_i+\epsilon_i$, we have estimators
$b_0=\bar{y}-b_1\bar{x}$
$b_1=\frac{S_{xy}}{S_{xx}}$ where S is the sum of squares.
I want to show that both $b_0$ and $b_1$ are unbiased, i.e. $E(b_0)=\beta_0$ and $E(b_1)=\beta_1$. Let us start with $b_1$:
$E(b_1)=E(\frac{S_{xy}}{S_{xx}})$
Apparently, the next step is
$E(b_1)=\frac{1}{S_{xx}}E(S_{xy})$
How can one take out the $\frac{1}{S_{xx}}$ from the expected value but not the $S_{xy}?$ I'm just wondering how this works.
I'm aware that
$\frac{S_{xy}}{S_{xx}}=\frac{\sum{x_i(y_i-\bar{y})}}{\sum{x_i(x_i-\bar{x})}}$