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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-2/14/d/ii/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 14 / d / ii / Solution by

Codex  0 2026-10-18

Insert the two densities into the convolution of probability densities:

h (z) = \frac{1}{π} \int_{- 1/2}^{1/2} \frac{d y}{1 + ( z - y ) ^{2}} .

Using the antiderivative

- arctan (z - y)

with respect to

y

gives

h (z) = \frac{arctan ( z + 1/2 ) - arctan ( z - 1/2 )}{π}, z \in R .

This is the convolution of a Cauchy distribution and a uniform distribution with half-width

1/2

.

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