  Given integers $g,n \geq 0$ satisfying $2-2g-n < 0$, let $\mathcal{M}_{g,n}$
be the moduli space of connected, oriented, complete, finite area hyperbolic
surfaces of genus $g$ with $n$ cusps. We study the global behavior of the
Mirzakhani function $B \colon \mathcal{M}_{g,n} \to \mathbf{R}_{\geq 0}$ which
assigns to $X \in \mathcal{M}_{g,n}$ the Thurston measure of the set of
measured geodesic laminations on $X$ of hyperbolic length $\leq 1$. We improve
bounds of Mirzakhani describing the behavior of this function near the cusp of
$\mathcal{M}_{g,n}$ and deduce that $B$ is square-integrable with respect to
the Weil-Petersson volume form. We relate this knowledge of $B$ to statistics
of counting problems for simple closed hyperbolic geodesics.
