Probability distributions with binomial moments

We prove that the binomial sequence [Formula: see text] is positive definite if and only if either p ≥ 1, -1 ≤ r ≤ p - 1 or p ≤ 0, p - 1 ≤ r ≤ 0 and that the Raney sequence [Formula: see text] is positive definite if and only if either p ≥ 1, 0 ≤ r ≤ p or p ≤ 0, p - 1 ≤ r ≤ 0 or else r = 0. The corresponding probability measures are denoted by ν(p, r) and μ(p, r) respectively. We prove that if p > 1 is rational and -1 < r ≤ p - 1 then the measure ν(p, r) is absolutely continuous and its density Vp,r(x) can be represented as Meijer G-function. In some cases Vp,r is an elementary function. We show that for p > 1 the measures ν(p,-1) and ν(p,0) are certain free convolution powers of the Bernoulli distribution.

The Number of Two Consecutive Successes in a Hoppe-Pólya Urn

In a sequence of independent Bernoulli trials the probability of success in the kth trial is pk = a / (a + b + k − 1). An explicit formula for the binomial moments of the number of two consecutive successes in the first n trials is obtained and some consequences of it are derived.