A binomial process is a special point process in probability theory.
Definition Let be a probability distribution and be a fixed natural number. Let be i.i.d. random variables with distribution , so for all .
Then the binomial process based on n and P is the random measure
where
Properties Name The name of a binomial process is derived from the fact that for all measurable sets the random variable follows a binomial distribution with parameters and :
Laplace-transform The Laplace transform of a binomial process is given by
for all positive measurable functions .
Intensity measure The intensity measure of a binomial process is given by
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 . Therefore, mixed binomial processes conditioned on are binomial process based on and .
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