Goal: Find, $\displaystyle Var(\hat B_0)$ and $\displaystyle Covar(\hat B_1, \hat B_0)$

$\displaystyle ($Note: $\displaystyle y_i = B_0 + x_i B_1 +e_i,$ with $\displaystyle e_i$ ~ $\displaystyle N(0,\sigma ^2)$ So, everything is fixed, but $\displaystyle e_i$ and $\displaystyle y_i.$ Now I found unbiased estimates $\displaystyle \hat B_0$ and $\displaystyle \hat B_1.$ Also, I found the variance of $\displaystyle \hat B_1.)$

$\displaystyle \hat B_0 = \bar y - {\bar x} \hat B_1$ and,

$\displaystyle \hat B_1 = \frac{\sum x_i y_i -{\bar y} \sum x_i}{\sum x_i ^2 - {\bar x} \sum x_i}$

Work:

$\displaystyle E[\hat B_0 ^2] = E[(\bar y - {\bar x}\hat B_1)^2] = E[\bar y ^2] - 2{\bar x}E[{\bar y}\hat B_1] + \bar x E[\hat B_1^2]$

Each part is easy enough, I just don't know how to start $\displaystyle E[{\bar y}\hat B_1]....?$

Lastly, how to approach $\displaystyle E[\hat B_0\hat B_1]....? $