First I need to minimize this just like I do with a regular L-S estimates (assuming Gauss-Markov) expect $\sum_{i=1}^n X_i=0$:
$$ \sum_i^n e_i^2=\sum_i^n(y_i-\beta_0+\beta_1x_i)^2 $$
I take the partial deriviates:
$$ -2\sum_i^n(y_i-\beta_0-\beta_1x_i) $$
$$ -2\sum_i^n(y_i-\beta_0-\beta_1x_i)x_i $$
Set equal to zero:
$$ \sum_i^n y_i-n\beta_0-\beta_1\sum_i^n x_i=0 \rightarrow \sum_i^n y_i-n\beta_0=0 \rightarrow n\beta_0=\sum_i^n y_i \rightarrow \beta_0=\bar{y} $$
$$ \sum_i^n x_iy_i-\beta_0x_i-\beta_1x_i^2=0 \rightarrow \beta_1\sum_i^nx_i^2=\sum_i^n x_iy_i \rightarrow \beta_1=\sum_i^n\frac{x_iy_i}{x_i^2}=\sum_i^n\frac{y_i}{x_i} $$
The expected values for these estimates are:
$$ E(\beta_0)=E(\bar(y))=E(\frac{\sum_i^n y_i}{n})=\frac{\sum_i^n E(y_i)}{n}=\frac{\sum_i^n \beta_0+\beta_1 \sum_i^nx_i}{n}=\beta_0+0=\beta_0 $$
$$ var(\beta_0)=var(\bar(y))=var(\frac{\sum_i^n y_i}{n})=\sum_i^n var(\frac{y_i}{n})=\sum_i^n \frac{var(y_i)}{n^2} = \frac{\sigma^2}{n} $$
I think my $\beta_0$ estimates are correct, but I feel like I did something wrong with $\beta_1$
I'd appreciate some guidance