In
probability theory, a birth process or a pure birth process is a special case of a
continuous-time Markov process
A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a ...
and a generalisation of a
Poisson process. It defines a continuous process which takes values in the
natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
and can only increase by one (a "birth") or remain unchanged. This is a type of
birth–death process
The birth–death process (or birth-and-death process) is a special case of continuous-time Markov process where the state transitions are of only two types: "births", which increase the state variable by one and "deaths", which decrease the state ...
with no deaths. The rate at which births occur is given by an
exponential random variable
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant averag ...
whose parameter depends only on the current value of the process
Definition
Birth rates definition
A birth process with birth rates
and initial value
is a minimal right-continuous process
such that
and the interarrival times
are independent
exponential random variable
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant averag ...
s with parameter
.
Infinitesimal definition
A birth process with rates
and initial value
is a process
such that:
*
*