Pure Birth Process
In probability theory, a birth process or a pure birth process is a special case of a continuous-time Markov process and a generalisation of a Poisson process. It defines a continuous process which takes values in the natural numbers and can only increase by one (a "birth") or remain unchanged. This is a type of birth–death process with no deaths. The rate at which births occur is given by an exponential random variable whose parameter depends only on the current value of the process Definition Birth rates definition A birth process with birth rates (\lambda_n, n\in \mathbb) and initial value k\in \mathbb is a minimal right-continuous process (X_t, t\ge 0) such that X_0=k and the interarrival times T_i = \inf\ - \inf\ are independent exponential random variables with parameter \lambda_i. Infinitesimal definition A birth process with rates (\lambda_n, n\in \mathbb) and initial value k\in \mathbb is a process (X_t, t\ge 0) such that: * X_0=k * \forall s,t\ge 0: s
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