Mazur, Rubin, and Stein have recently formulated a series of conjectures
about statistical properties of modular symbols in order to understand
central values of twists of elliptic curve L-functions. Two of these conjectures
relate to the asymptotic growth of the first and second moments of the modular
symbols. We prove these on average by using analytic properties of Eisenstein
series twisted by modular symbols. Another of their conjectures predicts the
Gaussian distribution of normalized modular symbols ordered according to
the size of the denominator of the cusps. We prove this conjecture in a refined
version that also allows restrictions on the location of the cusps.