I want to use mathematica to symbolically solve the minimization problem in simple linear regression:
$${\text{Find }}\text{arg}\min _{\alpha ,\,\beta }Q(\alpha ,\beta ),\qquad {\text{for }}Q(\alpha ,\beta ) =\sum _{i=1}^{n}(y_{i}-\alpha -\beta x_{i})^{2},$$
where $y_i$, $x_i$, and $n$ are symbolic but not specific numbers.The expected answer would be something like
$$\begin{align} \hat {\beta }&={\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}},\\ {\hat {\alpha }}&={\bar {y}}-{\hat {\beta }}\,{\bar {x}} \end{align}$$
Can someone show some code example for doing this? Thanks in advance!
