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{cases}1, &\text{if }X_i\in A,\\ 0, &\text{otherwise}.\end{cases}

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 \operatorname{Bin}(n,P(A)).

Laplace-transform

The Laplace transform of a binomial process is given by

\mathcal L_{P,n}(f)= \left[ \int \exp(-f(x)) \mathrm P(dx) \right]^n

for all positive measurable functions f.

Intensity measure

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

\operatorname{E}\xi =n P.

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.

Literature