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A Poisson process, one of a variety of things named after the 18th- and 19th-century French mathematician
Siméon-Denis Poisson, is a stochastic process 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
"arrivals" or "occurrences" in such a way that
- The number of arrivals in one interval of time or region in space and the number of arrivals in another disjoint
(non-overlapping) interval of time or region in space are independent random variables.
Technically, and perhaps more precisely, one should say each set of finite measure is assigned such a Poisson-distributed random variable.
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, λt. 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
-
-
- 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. This is a 2-dimensional Poisson process.
- Astonomers 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. This is a 3-dimensional Poisson
process.
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 number N(t) of arrivals before time t distribution has
a Poisson distribution with expected value
λt.
Let Xt be the number of arrivals before time t. Let Tx be
the time of the xth 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 constants, i.e., non-random quantities.) The random
variable Xt has a discrete probability distribution -- a Poisson distribution -- and the
random variable Tx 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
xth arrival is more than t. In symbols, the event [ Xt < x ] occurs if
and only if the event [ Tx > t ]. Consequently the probabilities of these events are the
same:
- P(Xt < x) = P(Tx >
t).
This fact plus knowledge of the Poisson distribution enables us to find the probability distribution of these continuous
random variables. In case 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(T1 > t) = P(Xt = 0) = e
- λt.
Consequently, the waiting time until the first arrival has a 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
-
This says that the conditional probability that we
need to wait, for example, more than another 10 seconds before the first arrival, given that the first arrival has not yet
happened after 30 seconds, is no different from the initial probability that we need to wait more than 10 seconds for the first
arrival. This is often misunderstood by students taking courses on probability: the fact that P(T1 > 40 |
T1 > 30) = P(T1 > 10) does not mean that the events T1
> 40 and T1 > 10 are independent. To summarize: "memorlessness" of the probability distribution of the
waiting time T1 until the first arrival means
-
It does not mean
-
(That would be independence. These two events are not independent.)
Characterization of Poisson processes
In its most general form, the only two conditions for a 1-dimensional process to be a (not necessarily homogeneous) Poisson
process are:
- Orderliness: which roughly means limΔ → 0 Pr[Xt + Δ −
Xt > 1 | Xt + Δ − Xt ≥ 1]
→ 0 which implies that events don't occur simultaneously (but is actually a stronger statement).
- Memorylessness (also called evolution without aftereffects): the number of arrivals occurring in any bounded interval
of time after time t is independent of the
number of arrivals occuring before time t.
These seemingly unrestrictive conditions actually impose a great deal of structure in the Poisson process. In particular, they
imply independent exponential (memoryless) interarrival times (with parameter λ for homogeneous processes). Because the
interarrival times are exponentially distributed,
the time between the 4th and 9th arrival (for instance) is distributed as the sum of exponential random variables (i.e. 5th order
gamma distribution). Also, these conditions imply that the
probability distribution of the number of events in the interval [a,b), which is also written as Xb
− Xa is Poisson-distributed, (with parameter λ(b − a) for homogeneous
processes).
This is a sample one-dimensional homogeneous Poisson process, Xt; not to be confused with a
density or distribution function.
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