  We show that $$ \sum_{n\neq
m}\frac{\mu(n)\mu(m)}{nm}E_{X}\left(\{nx\}\{mx\}\right)=-\frac{9}{2\pi^{2}}+O\left(\frac{1}{X}\right),
$$ where $x$ is uniformly distributed in $[0,X]$ with $X\in \mathbb{N}$,
$E_{X}(.)$ denotes the expected value, $\mu(.)$ denotes the M\"obius function,
and $\{.\}$ denotes the fractional part function.
