Binomial process

A binomial process is a special point process in probability theory.

Definition

Let $P$ be a probability distribution and $n$ be a fixed natural number. Let $X_1,X_2,\dots ,X_n$ be i.i.d. random variables with distribution $P$, so $X_i\sim P$ for all $i\in \{1,2,\dots ,n\}$.

Then the binomial process based on n and P is the random measure

$\xi =\sum _{i=1}^n{\delta }_{X_i},$

where ${\delta }_{X_i(A)}=\{\begin{array}{cc}1,& \text{if }{X}_i\in A,\\ 0,& \text{otherwise}.\end{array}$

Properties

Name

The name of a binomial process is derived from the fact that for all measurable sets $A$ the random variable $\xi (A)$ follows a binomial distribution with parameters $P(A)$ and $n$:

$\xi (A)\sim \mathrm{Bin}(n,P(A)).$

Laplace-transform

The Laplace transform of a binomial process is given by

${ℒ}_{P,n}(f)={[\int \exp (-f(x))\mathrm{P}(dx)]}^n$

for all positive measurable functions $f$.

Intensity measure

The intensity measure $\mathrm{E}\xi$ of a binomial process $\xi$ is given by

$\mathrm{E}\xi =nP.$

Generalizations

A generalization of binomial processes are mixed binomial processes. In these point processes, the number of points is not deterministic like it is with binomial processes, but is determined by a random variable $K$. Therefore mixed binomial processes conditioned on $K=n$ are binomial process based on $n$ and $P$.