  We study several probability distributions relevant to the avalanche dynamics
of elastic interfaces driven on a random substrate: The distribution of size,
duration, lateral extension or area, as well as velocities. Results from the
functional renormalization group and scaling relations involving two
independent exponents, roughness $\zeta$, and dynamics $z$, are confronted to
high-precision numerical simulations of an elastic line with short-range
elasticity, i.e. of internal dimension $d=1$. The latter are based on a novel
stochastic algorithm which generates its disorder on the fly. Its precision
grows linearly in the time-discretization step, and it is parallelizable. Our
results show good agreement between theory and numerics, both for the critical
exponents as for the scaling functions. In particular, the prediction ${\sf a}
= 2 - \frac{2}{d+ \zeta - z}$ for the velocity exponent is confirmed with good
accuracy.
