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Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S3 / 2 / b / Solution 2026-10-25

The probability density is even, so its first moment vanishes. Its second moment follows by integrating the polynomial:

⟨ x^{2} ⟩ = A^{2} [\frac{2 a ^{7}}{3} - \frac{4 a ^{7}}{5} + \frac{2 a ^{7}}{7}] .

Using the normalization from part a gives

⟨ x ⟩ = 0, ⟨ x^{2} ⟩ = \frac{a ^{2}}{7} .

The variance in position is consequently

a^{2} /7

.

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