A '''binomial process''' is a special point process in probability theory.
== Definition == Let <math> P </math> be a probability distribution and <math> n </math> be a fixed natural number. Let <math> X_1, X_2, \dots, X_n </math> be i.i.d. random variables with distribution <math> P </math>, so <math> X_i \sim P </math> for all <math> i \in \{1, 2, \dots, n \}</math>.
Then the binomial process based on ''n'' and ''P'' is the random measure
: <math> \xi= \sum_{i=1}^n \delta_{X_i}, </math> where <math>\delta_{X_i(A)}=\begin{cases}1, &\text{if }X_i\in A,\\ 0, &\text{otherwise}.\end{cases}</math>
== Properties == === Name === The name of a binomial process is derived from the fact that for all measurable sets <math> A </math> the random variable <math> \xi(A) </math> follows a binomial distribution with parameters <math> P(A) </math> and <math> n </math>:
:<math> \xi(A) \sim \operatorname{Bin}(n,P(A)).</math>
=== Laplace-transform === The Laplace transform of a binomial process is given by :<math> \mathcal L_{P,n}(f)= \left[ \int \exp(-f(x)) \mathrm P(dx) \right]^n </math>
for all positive measurable functions <math> f </math>.
=== Intensity measure === The intensity measure <math> \operatorname{E}\xi </math> of a binomial process <math> \xi </math> is given by
:<math> \operatorname{E}\xi =n P.</math>
== 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 <math> K </math>. Therefore, mixed binomial processes conditioned on <math> K=n </math> are binomial process based on <math> n </math> and <math> P </math>.
== Literature == *{{cite book |last1=Kallenberg |first1=Olav |author-link1=Olav Kallenberg |year=2017 |title=Random Measures, Theory and Applications|location= Switzerland |publisher=Springer |doi= 10.1007/978-3-319-41598-7|isbn=978-3-319-41596-3}}
Category:Point processes