  A chord diagram is a set of $n$ chords joining distinct endpoints on a
circle. We study the asymptotic distributions of the number of crossings and
the number of simple chords in a random chord diagram. Using size-bias coupling
and Stein's method, we obtain bounds on the Wasserstein distance between the
distribution of the number of crossings and a standard normal. We also give
upper and lower bounds of the same order on the total variation distance
between the distribution of the number of simple chords and a Poisson random
variable.
