A mixed binomial process is a special point process in probability theory.
They naturally arise from restrictions of (mixed) Poisson processes bounded intervals.
random variables with distribution
be a random variable taking a.s. (almost surely) values in
denote the Dirac measure on the point
Then a random measure
is called a mixed binomial process iff it has a representation as This is equivalent to
being a binomial process based on
, a mixed Binomial processe has the Laplace transform for any positive, measurable function
and a bounded measurable set
as Mixed binomial processes are stable under restrictions in the sense that if
is a mixed binomial process based on
is a mixed binomial process based on and some random variable
is a Poisson process or a mixed Poisson process, then
is a mixed binomial process.
[2] Poisson-type random measures are a family of three random counting measures which are closed under restriction to a subspace, i.e. closed under thinning, that are examples of mixed binomial processes.
They are the only distributions in the canonical non-negative power series family of distributions to possess this property and include the Poisson distribution, negative binomial distribution, and binomial distribution.
Poisson-type (PT) random measures include the Poisson random measure, negative binomial random measure, and binomial random measure.