Expected value of the smallest denominator in a random interval of fixed radius

We compute the probability mass function of the random variable which returns the smallest denominator of a reduced fraction in a randomly chosen real interval of radius $\delta \mathrm{/}2$. As an application, we prove that the expected value of the smallest denominator is asymptotic, as $\delta \to 0$, to $(16\mathrm{/}{\pi }^2){\delta }^{-1\mathrm{/}2}\mathrm{.}$