 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2011/p/p3/d/solution

By part (c), the best degree-at-most-two polynomial is odd, so it must be

p_{2} (x) = b x

x

gives

b = \frac{( x ^{2011} , x )}{( x , x )} = \frac{\int _{- 1}^{1} x ^{2014} d x}{\int _{- 1}^{1} x ^{4} d x} = \frac{2/2015}{2/5} = \frac{1}{403} .

Therefore

p_{2} (x) = \frac{x}{403} .

Its error is automatically orthogonal to the even basis functions

1, x^{2}

by parity, and the chosen coefficient makes it orthogonal to

x

, so part (b) verifies optimality.

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