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 $\delta _{X_{i}(A)}={\begin{cases}1,&{\text{if }}X_{i}\in A,\\0,&{\text{otherwise}}.\end{cases}}$

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 =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 $P$ .

Literature

Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.

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