Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 18 / b

Each prediction is a linear combination of the constant parameter and the coefficient of $x_i$. Thus the design matrix isIts $i$th row multiplied by the parameter vector $(\alpha ,\beta {)}^{\mathsf{T}}$ gives the required prediction $\alpha +\beta x_i$.

Compute the two sides of the normal equations for linear least squares and divide by $n$:The first equation gives $\alpha =\overline{y}-\beta \overline{x}$. Substitute this into the second and solve:These simple linear regression coefficients are unique provided the observed distances are not all equal, sinceThe condition $n\ge 2$ alone does not guarantee this. If all $x_i=x_0$, the design matrix has matrix rank one and the valid alternative is the familyIn this rank-deficient ordinary least squares case, the fitted predictions are uniquely $\overline{y}$ but the two parameters cannot be separately determined.