Derivation of Ordinary Least-Squares Slope Estimate

Question

I understand the reasoning behind the fact that $\hat{\beta}_1 \sim N(\beta_1,\frac{\sigma^2}{SSX})$. However, in trying to prove the full formula for estimating $\hat{\beta}_1$, I am having issues/don't know the reasoning behind this part of the formula:

$\sum_{i=1}^n (X_i - \bar{X})(Y_i - \bar{Y})$ = $\sum_{i=1}^n (X_i - \bar{X})Y_i$

Can anyone help me out? Thanks!

Answer 1

\begin{align} \sum (X_i - \bar{X})(Y_i - \bar{Y}) &= \sum (X_i - \bar{X})Y_i - \sum (X_i - \bar{X})\bar{Y} \\ &=\sum (X_i - \bar{X})Y_i - \sum X_i\bar{Y} + n\bar{X}\bar{Y} \\ &= \sum (X_i - \bar{X})Y_i - n\bar{X}\bar{Y} + n\bar{X}\bar{Y}\\ &= \sum (X_i - \bar{X})Y_i \end{align}