Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2018 / Paper 1 / 16 / d

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- Solution d

Solution

Differentiate the cumulative distribution function using the chain rule:

f (y) = F^{'} (y) = \frac{1}{π x _{0}} \frac{1}{1 + ( y / x _{0} ) ^{2}} = \frac{x _{0}}{π ( x _{0}^{2} + y ^{2} )} .

This is a Cauchy distribution with location zero and scale

x_{0}

. The probability density function is nonnegative and integrates to

F (\infty) - F (- \infty) = 1

.

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