A discrete random variable X  is said to have a Borel distribution[1][2] with parameter μ ∈ [0,1] if the probability mass function of X is given by for n = 1, 2, 3 ....

If a Galton–Watson branching process has common offspring distribution Poisson with mean μ, then the total number of individuals in the branching process has Borel distribution with parameter μ.

Since the mth generation of the branching process has mean size μm − 1, the mean of X  is In an M/D/1 queue with arrival rate μ and common service time 1, the distribution of a typical busy period of the queue is Borel with parameter μ.

[6] If Pμ(n) is the probability mass function of a Borel(μ) random variable, then the mass function P∗μ(n) of a sized-biased sample from the distribution (i.e. the mass function proportional to nPμ(n) ) is given by Aldous and Pitman [7] show that In words, this says that a Borel(μ) random variable has the same distribution as a size-biased Borel(μU) random variable, where U has the uniform distribution on [0,1].

Generalizing the random walk correspondence given above for k = 1,[4][5] where Sn has Poisson distribution with mean nμ.