We describe a probability distribution on isomorphism classes of principally
quasi-polarized p-divisible groups over a finite field k of characteristic p
which can reasonably be thought of as "uniform distribution," and we compute
the distribution of various statistics (p-corank, a-number, etc.) of
p-divisible groups drawn from this distribution. It is then natural to ask to
what extent the p-divisible groups attached to a randomly chosen hyperelliptic
curve (resp. curve, resp. abelian variety) over k are uniformly distributed in
this sense.