• General Equation of Multiple Linear Regressor

    $$ E[Y|X] = f(X)= \beta_0 +\beta_1X_1+...+\beta_pX_p+\epsilon $$

  • Error in Multiple Linear Regressor

    $$ e_i = y_i-\^{y}_i $$

    $$ RSS = e^2_1+e^2_2+....+e^2_n=E[(Y-\^{Y})^2 |X] $$

  • Confidence Intervals:

    More Generalized Confidence Intervals

    • $n$ represents the degree of freedom

    • $\alpha$ represents the p value.

    • $p$ represents number of features.

      $$ [\;\hat\beta_i\pm t_{n-p-1,\alpha/2}*SE(\hat\beta_i)\;] $$

  • Hypothesis Testing:

    • Assume

      $$ H_0: \beta_i = 0 \newline H_A:\beta_i\ne0 $$

      • since $\beta_i$ = 0
      • We do the following to test

      $$ H_0: \beta_i = \beta \newline H_A:\beta_i\ne \beta $$

    • We get that

      $$ t_{obs} = \frac{\hat\beta_i - \beta}{SE)\hat\beta_i} ;\; given \;\beta_i=\beta\; ;usually \;\beta=0 $$

    • Reject if:

      $$ |t_{obs}| > t_{n-p-1,\alpha/2} $$

      • if p $\uarr$ the n $\darr$ tail thickens