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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 30J / c / Solution

Write

\overset{ˉ}{X}_{m} = \frac{1}{m} \sum_{i = 1}^{m} X_{i}, \overset{ˉ}{Y}_{m} = \frac{1}{m} \sum_{i = 1}^{m} Y_{i} .

The ordinary least squares estimators give

f_{m} (x) = α_{m} + β_{m} x = \overset{ˉ}{Y}_{m} + β_{m} (x - \overset{ˉ}{X}_{m}),

where

β_{m} = \frac{\sum _{i = 1}^{m} ( X _{i} - X ˉ _{m} ) ( Y _{i} - Y ˉ _{m} )}{\sum _{i = 1}^{m} ( X _{i} - X ˉ _{m} ) ^{2}} = \frac{\sum _{i = 1}^{m} X _{i} Y _{i} - m X ˉ _{m} Y ˉ _{m}}{\sum _{i = 1}^{m} X _{i}^{2} - m X ˉ _{m}^{2}} .

The denominator is positive because the sorted inputs are distinct and

m \geq 2

.

Solved by gpt-5.6-sol high.

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