Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 4 / 17H / a

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- Solution a

Solution

Let

S_{xx} = \sum_{i = 1}^{n} (x_{i} - \overset{x}{ˉ})^{2}, S_{x y} = \sum_{i = 1}^{n} (x_{i} - \overset{x}{ˉ}) (Y_{i} - \overset{ˉ}{Y}) .

Differentiating the Gaussian log-likelihood, equivalently minimizing the residual sum of squares, gives the ordinary least squares estimators

\hat{β} = \frac{S _{x y}}{S _{xx}}, \overset{α}{^} = \overset{ˉ}{Y} - \hat{β} \overset{x}{ˉ},

assuming

S_{xx} > 0

.

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