World Library  
Flag as Inappropriate
Email this Article

Borel distribution

Article Id: WHEBN0040323986
Reproduction Date:

Title: Borel distribution  
Author: World Heritage Encyclopedia
Language: English
Subject: M/D/1 queue, Multivariate Pareto distribution, Hyper-Erlang distribution, Dyadic distribution, Chernoff's distribution
Publisher: World Heritage Encyclopedia

Borel distribution

Borel distribution
Parameters \mu \in [0,1]
Support n \in \{1, 2, 3,\ldots\}
pmf \frac{e^{-\mu n}(\mu n)^{n-1}}{n!}
Mean \frac{1}{1-\mu}
Variance \frac{\mu}{(1-\mu)^3}

The Borel distribution is a discrete probability distribution, arising in contexts including branching processes and queueing theory.

If the number of offspring that an organism has is Poisson-distributed, and if the average number of offspring of each organism is no bigger than 1, then the descendants of each individual will ultimately become extinct. The number of descendants that an individual ultimately has in that situation is a random variable distributed according to a Borel distribution.


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

P_\mu(n)= \Pr(X=n)= \frac{e^{-\mu n}(\mu n)^{n-1}}{n!}

for n = 1, 2, 3 ....

Derivation and branching process interpretation

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.

Let X  be the total number of individuals in a Galton–Watson branching process. Then a correspondence between the total size of the branching process and a hitting time for an associated random walk[3][4][5] gives


where Sn = Y1 + … + Yn, and Y1 … Yn are independent identically distributed random variables whose common distribution is the offspring distribution of the branching process. In the case where this common distribution is Poisson with mean μ, the random variable Sn has Poisson distribution with mean μn, leading to the mass function of the Borel distribution given above.

Since the mth generation of the branching process has mean size μm − 1, the mean of X  is

1+\mu+\mu^2+\cdots = \frac{1}{1-\mu}.

Queueing theory interpretation

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
(n) of a sized-biased sample from the distribution (i.e. the mass function proportional to nPμ(n) ) is given by

P_\mu^*(n)=(1-\mu)\frac{e^{-\mu n}(\mu n)^{n-1}}{(n-1)!}.

Aldous and Pitman [7] show that

P_\mu(n)=\frac{1}{\mu}\int_0^{\mu}P_\lambda^*(n) \, d\lambda.

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].

This relation leads to various useful formulas, including

E(1/X) = 1-\mu/2.

Borel–Tanner distribution

The Borel–Tanner distribution generalizes the Borel distribution. Let k be a positive integer. If X1X2,  … Xk are independent and each has Borel distribution with parameter μ, then their sum W = X1 + X2 + … + Xk is said to have Borel–Tanner distribution with parameters μ and k. [2][6][8] This gives the distribution of the total number of individuals in a Poisson–Galton–Watson process starting with k individuals in the first generation, or of the time taken for an M/D/1 queue to empty starting with k jobs in the queue. The case k = 1 is simply the Borel distribution above.

Generalizing the random walk correspondence given above for k = 1,[4][5]


where Sn has Poisson distribution with mean . As a result the probability mass function is given by

\Pr(W=n)=\frac{k}{n}\frac{e^{-\mu n}(\mu n)^{n-k}}{(n-k)!}

for n = kk + 1, ... .


  1. ^
  2. ^ a b
  3. ^
  4. ^ a b
  5. ^ a b
  6. ^ a b
  7. ^
  8. ^

External links

  • Borel-Tanner distribution in Mathematica.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.