Given a hyperbolic surface X and any closed geodesic \(\gamma \) on X, Mirzakhani provided an asymptotic equivalent of the number \(N_\gamma (a)(X)\) of geodesics of length at most a in the mapping class group orbit of \(\gamma \) , when \(a\rightarrow \infty \) . After integrating this result on the moduli space, she obtained a similar asymptotic equivalent of the Weil-Petersson expectation \(\mathbb {E}[N_\gamma (a)]\) . In this paper we use Mirzakhani’s integration formula in order to integrate random variables on the moduli space whose definition involves lengths of eight-shaped geodesics, and we compute the asymptotic behavior of the density of these geometric random variables by using tools developed by Anantharaman and Monk. As a consequence, we improve the asymptotic expansion of the average counting function \(a\mapsto \mathbb {E}[N_\gamma (a)]\) for geodesics \(\gamma \) with exactly one self-intersection and we provide an explicit computation of the leading term of this expansion when \(\gamma \) is an eight-shaped geodesic on a n-holed sphere.
