Least square estimator of β in linear regression
Assume $$ Y_i = \beta_o + \beta_1X_i+\varepsilon_i $$ , for given $n$ observerd data $(x_i, Y_i)$, $\forall i=1～n$ Note that: $$ Y_i \mid X_i=x_i ~ N(\beta_o+\beta_1X_1, \sigma^2) $$ $\therefore$ $$ E_{Y \mid X}[Y_i \mid X_i =x_i]=\beta_o+\beta_1X_1$$ In vector notation: $$ Y_i = x^T_i\beta + \varepsilon_i $$ where $ x_i=(1, X_1, …, X_n)^T $ And for $ Y = (Y_1, …, Y_n)^T $, We have: $$ Y = X\beta + \varepsilon $$ ...