For the simple linear regression model, show that the elements of the hat matrix are [h_{i j}=frac{1}{n}+frac{left(x_{i}-bar{x} ight)left(x_{j}-bar{x} ight)}{S_{x x}} text { and } h_{i

For the simple linear regression model, show that the elements of the hat matrix are

\[h_{i j}=\frac{1}{n}+\frac{\left(x_{i}-\bar{x}\right)\left(x_{j}-\bar{x}\right)}{S_{x x}} \text { and } h_{i i}=\frac{1}{n}+\frac{\left(x_{i}-\bar{x}\right)^{2}}{S_{x x}}\]

Discuss the behavior of these quantities as $x_{i}$ moves farther from $\bar{x}$,