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  • The number of events occurring in two Disjoint (non-overlapping) subintervals are Independent random variables.


  • The probability of the number of events in some subinterval [t,t+ au] is given by

    • : P [(N(t+ au) - N(t)) = k] = rac{e^{-\lambda au} (\lambda au)^k}{k!} \qquad k= 0,1,\ldots

where the positive number λ is a fixed parameter, known as the rate parameter. In words, this means that the random variable N(t+ au) - N(t), giving the number of occurrences in the time interval [t,t+ au], has a Poisson Distribution with parameter \lambda au.

More generally, a Poisson process is one that assigns to each bounded interval of time or to each bounded region in some space (for example, a Euclidean plane or a 3-dimensional Euclidean space) a random number of events in such a way that

  • The number of events in one interval of time or region in space and the number of events in another disjoint (''non-overlapping'') interval of time or region in space are Independent random variables; and


  • The number of events in each interval of time or region in space is a Random Variable with a Poisson Distribution . ( Technically, and perhaps more precisely, one should say each set of finite Measure is assigned such a Poisson-distributed random variable.)


The Poisson process is one of the most well-known Lévy Process es. (Time-homogeneous) Poisson processes are also examples of (time-homogeneous) Continuous-time Markov Process es. A time-homogeneous, one-dimensional Poisson process is a pure-birth process, the simplest example of a Birth-death Process .


EXAMPLES


  • The number of telephone calls arriving at a switchboard during any specified time interval may have a Poisson distribution, and the number of calls arriving during one time interval may be Statistically Independent of the number of calls arriving during any other non-overlapping time interval. This is a one-dimensional Poisson process. In simple models, one may assume a constant average rate of arrival, e.g., λ = 12.3 calls per minute. In that case, the Expected Value of the number of calls in any time interval is that rate times the amount of time, λ''t''. In messier and more realistic problems, one uses a non-constant rate function λ(''t''). In that case, the expected value of the number of calls between time ''a'' and time ''b'' is


::\int_a^b \lambda(t)\,dt.

  • The number of photons hitting a photodetector during a specified time interval may follow a Poisson distribution.


  • The number of bombs falling on a specified area of London in the early days of the Second World War may be a random variable with a Poisson distribution, and the number of bombs falling on two areas of the city that do not overlap may be statistically independent. The number of bombs observed to have fallen within an area ''A'' is a 2-dimensional Poisson process over the space defined by the area ''A''.


  • Astronomers may treat the number of stars in a given volume of space as a random variable with a Poisson distribution, and the numbers of stars in any two or more non-overlapping regions as statistically independent. The number of stars observed within some volume ''V'' is a 3-dimensional Poisson process over the space defined by the volume ''V''.



1-DIMENSIONAL POISSON PROCESSES


A 1-dimensional Poisson process on the interval from 0 to ∞ (essentially this means that the clock starts at time 0; that is when we begin counting) may thus be viewed as an Integer -valued nondecreasing random function of time ''N''(''t'') that counts the number of "arrivals" before time ''t''. Just as a Poisson random variable is characterized by its scalar parameter λ, a Poisson process is characterized by its rate function λ(''t''), which is the Expected number of "events" or "arrivals" that occur per unit time. A ''homogeneous'' Poisson process has a constant rate function λ(''t'') = λ. If the rate remains constant, then the distribution of the number ''N''(''t'') of arrivals before time ''t'' follows a Poisson Distribution with expected value λ''t''.

Let ''X''''t'' be the number of arrivals before time ''t''. Let ''T''''x'' be the time of the ''x''th arrival, for ''x'' = 1, 2, 3, ... . (We are using capital ''X'' and capital ''T'' for random variables, and lower-case ''x'' and lower-case ''t'' for non-random quantities.) The random variable ''X''''t'' has a ''discrete'' Probability Distribution -- a Poisson distribution -- and the random variable ''T''''x'' has a ''continuous'' probability distribution.

Clearly the number of arrivals before time ''t'' is less than ''x'' if and only if the waiting time until the ''x''th arrival is more than ''t''. In symbols, the event ''X''''t'' < ''x'' occurs if and only if the event ''T''''x'' > ''t'' . Consequently the probabilities of these events are the same:

:P(X_tt).

This fact plus knowledge of the Poisson distribution enables us to find the probability distribution of these continuous random variables. In the case where the rate, i.e., the expected number of arrivals per unit time, remains constant, this is fairly simple. In particular, consider the waiting time until the first arrival. Clearly that time is more than ''t'' if and only if the number of arrivals before time ''t'' is a 0. If the rate is λ arrivals per unit time, then we have

:P(T_1>t)=P(X_t=0)=e^{-\lambda t}.

Consequently, the waiting time until the first arrival has an Exponential Distribution . This exponential distribution has expected value 1/λ. In other words, if the average rate of arrivals is, for example 6 per minute, then the average waiting time until the first arrival is (unsurprisingly) 1/6 minute. This exponential distribution is Memoryless , i.e. we have

:P(T_1>t+s \mid T_1>t)=P(T_1>s).