Past exam of the Probability, statistics and options II course of the University of Oxford / 2008 / P / 3 / c

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- Solution c

Solution

Write

q (x) = a + b x

. Orthogonality of

x^{2} - q

to

1, x

gives

(lo g 3 2 24) (b a) = (26/3 4) .

The entries are obtained by integrating

1/ x, 1, x, x^{2}

over the given interval. Subtracting the second equation divided by two from the first gives

(lo g 3 - 1) a = - 1/3

. Thus

q (x) = - \frac{1}{3 ( lo g 3 - 1 )} + (\frac{13}{6} + \frac{1}{6 ( lo g 3 - 1 )}) x .

The Gram matrix is positive definite, with determinant

4 (lo g 3 - 1) > 0

, so this orthogonal projection is the unique best approximation.

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