When a uniform ratio rounds to a square

HardProbability Random Variables~10m

Let $X, Y \sim \text{Uniform}(0,1)$ be independent. Compute the probability that $\left\lceil \dfrac{Y}{X} \right\rceil$ is a perfect square. The answer has the form $q - \dfrac{\pi^2}{a}$ for a rational number $q$ and an integer $a$ . What is $a\cdot q$ ?

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First find the CDF of $R = Y/X$ : it is $t/2$ for $t\le1$ and $1 - 1/(2t)$ for $t\ge1$ . Then $P(\lceil R\rceil=k)$ is a difference of CDF values.
For $k\ge2$ , $P(\lceil R\rceil=k)=\tfrac{1}{2k(k-1)}$ . Sum over $k=m^2$ ; split into a telescoping series and the Basel sum $\sum 1/m^2 = \pi^2/6$ . Don't forget the $k=1$ term.

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Asked at: Jane Street, Citadel

When a uniform ratio rounds to a square

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