[\textrm{Var }\hat{\beta}{0}=\frac{\sigma^{2}\sum{i=1}^{n}x{i}^{2}}{n\sum{i=1}^{n}x{i}^{2}-\left(\sum{i=1}^{n}x{i}\right)^{2}}][\textrm{Var }\hat{\beta}{1}=\frac{n\sigma^{2}}{n\sum{i=1}^{n}x{i}^{2}-\left(\sum{i=1}^{n}x{i}\right)^{2}}][\textrm{Cov}(\hat{\beta}{0},\hat{\beta}{1})=\frac{-\sigma^{2}\sum{i=1}^{n}x{i}}{n\sum{i=1}^{n}x{i}^{2}-\left(\sum{i=1}^{n}x{i}\right)^{2}}]

$\textrm{Var }\hat{\beta}_{0}=\frac{\sigma^{2}\sum_{i=1}^{n}x_{i}^{2}}{n\sum_{i=1}^{n}x_{i}^{2}-\left(\sum_{i=1}^{n}x_{i}\right)^{2}}$ $\textrm{Var }\hat{\beta}_{1}=\frac{n\sigma^{2}}{n\sum_{i=1}^{n}x_{i}^{2}-\left(\sum_{i=1}^{n}x_{i}\right)^{2}}$ $\textrm{Cov}(\hat{\beta}_{0},\hat{\beta}_{1})=\frac{-\sigma^{2}\sum_{i=1}^{n}x_{i}}{n\sum_{i=1}^{n}x_{i}^{2}-\left(\sum_{i=1}^{n}x_{i}\right)^{2}}$

Bevis för $\hat{\beta}_1$

📜

\[\hat{\beta}_{1}=\frac{\sum(x_{i}-\overline{x})(y_{i}-\overline{y})}{\sum(x_{i}-\overline{x})^{2}}=\frac{\sum(x_{i}y_{i}-x_{i}\overline{y}-\overline{x}y_{i}+\overline{x}\overline{y})}{\sum(x_{i}-\overline{x})^{2}}=\frac{\sum(x_{i}-\overline{x})y_{i}-\overline{y}\overbrace{\sum(x_{i}-\overline{x})}^{=0}}{\sum(x_{i}-\overline{x})^{2}}=\frac{\sum(x_{i}-\overline{x})y_{i}}{\sum(x_{i}-\overline{x})^{2}}\]

\[\textrm{Var }\hat{\beta}_{1}=\frac{\sum(x_{i}-\overline{x})^{2}\overbrace{\sigma^{2}}^{=\textrm{Var }y_{i}}}{\left(\sum(x_{i}-\overline{x})^{2}\right)^{2}}=\frac{\sigma^{2}}{\sum(x_{i}-\overline{x})^{2}}=\frac{n\sigma^{2}}{n\sum x_{i}^{2}-\left(\sum x_{i}\right)^{2}}\]

Under the
standard statistical model