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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, random graphs and queueing theory.

Definition

A discrete random variable X  is said to have a Borel distribution with parameter $\mu \in [0,1]$ 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 G is a Galton-Watson branching process whose common offspring distribution is 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 gives

$\Pr(X=n)={\frac {1}{n}}\Pr(S_{n}=n-1)$

where $S_{n}=Y_{1}+...+Y_{n}$, and $Y_{1},...,Y_{n}$ 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 $S_{n}$ 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 $\mu ^{m-1}$, the mean of X  is $1+\mu +\mu ^{2}+...=1/(1-\mu )$.

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.

Properties

If $P_{\mu }(n)$ is the probability mass function of a Borel(μ) random variable, then the mass function $P_{\mu }^{*}(n)$ of a sized-biased sample from the distribution (i.e. the mass function proportional to $nP_{\mu }(n)$) is given by

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

Aldous and Pitman 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

Let $k\in \{1,2,3,...\}$. If $X_{1},X_{2},...,X_{k}$ are independent and each have Borel distribution with parameter μ, then their sum $W=X_{1}+X_{2}+...+X_{k}$ is said to have Borel-Tanner distribution with parameters μ and k. 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 representation given above for k=1,

$\Pr(W=n)={\frac {k}{n}}\Pr(S_{n}=n-k)$

where $S_{n}$ has Poisson distribution with mean nμ. 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=k, k+1, ....

External links

Borel-Tanner distribution in Mathematica.