The Poisson Process

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1. Priliminary Definitions

Def : A stochastic process is a collection of random variable (RV) indexed by time {X(t), t in T}.

  • If T is continuous set, the process is a continuous time stochastic process (e.g., Poisson Process)
  • If T is countable, then the process is a discrete time stochastic process (e.g., Markov chain)
Def. A counting process is a stochastic process {N(t); t geq 0} such that
  • N(t) in {0,1,cdots, 2} (that is, N(t) is non-negative integer)
  • If s < t, the N(s) leq N(t) ( that is, N(t) is non-decreasing in t)
  • For s<t, N(t) - N(s) is the number of events occurring in the time interval (s,t].

Def: A counting process has stationary increments if the distribution of the number of events in an interval depends on the length of the interval, but not on the starting point of the interval. That is, P(N(s+t) - N(s) = n) does not depend on s. Intuitively, the interval can be “slide” around without changing its stochastic nature.

2. Definition of Poisson Process

Definition 1: A Poisson process is a counting process {N(t); t geq 0} with rate lambda >0, if:

  • N(0) = 0;
  • The Process has independent increments
  • The number of events in any interval of length t is a Poisson RV with mean lambda t.
That is, for all s,t geq 0 and n = 0,1,2, cdots
P(N(s+t) - N(s) = n) = e^{-lambda t} (lambda t)^{n}/n!.

Definition 2: A Poisson process is a counting process {N(t); t geq 0} with rate lambda >0 if

  1. N(0) = 0
  2. The process has stationary increments
  3. The process has independent increments
  4. Pr{N(h) =1} = lambda h + o(h) (# of events approximately proportional to the length of interval)
  5. Pr{N(h) geq 2} = o(h) (can’t have 2 or more events at the same time — “orderliness”)
Eliminating of individual assumptions yields variations on the Poisson process
  • Eliminate assumption 2, Non-stationary Poisson Process
  • Eliminate assumption 3, Mixture of Poisson Process (choose lambda randomly, then run a Poisson process)
  • Eliminate assumption 5, compound Poisson Process
Definition 3: A Poisson process with rate lambda is a counting process such that times between events are i.i.d distribution exp(lambda)

3. Conditional Distribution of Event Times


4. Example

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