# Talk:Poisson process

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## Early discussion

How come there is no mention of how to *sample* a Poisson process (of any kind)? —Preceding unsigned comment added by 171.64.68.49 (talk) 03:17, 17 December 2009 (UTC)

I've taken quite a few liberties on my last set of edits so a justification would probably be in order. The major change I made was to treat spatial processes as an extension of Poisson processes instead of treating temporal Poisson processes as a special case of multi-dimensional Poisson processes. I believe this change is justified, as a Poisson process is after all a stochastic process which is tradiationally a random function of time. I believe multidimensional Poisson processes are worth mentioning, but perhaps would warrant their own article as there are characteristic differences about them that aren't done justice in this article. For example, the number of events in a given time interval is given as N(b)-N(a), but the formulation for the number of events in a (not necessarily rectangular) region of space based on the function N(x,y) is not nearly as straightforward. Also, the memorylessness property of multidimensional P.P. is not well motivated. (I omitted further discussion on spatial P.P. to keep from pushing this important concept of memorylessness even further down the article.) Until and unless this separation proposition reaches some consensus, I've left spatial P.P. as a subsection. (See also comments on the bomb analogy below.) Willem 22:23, 7 October 2006 (UTC)

I felt the introduction was a bit hard to understand for somebody with no prior knowledge (see also the comments further below on this talk page). I hope I was able to improve this point. I agree with the separation if we have something like Poisson process (real line) and Poisson process (spatial) with a redirect from Poisson process to the first. Being a probabilist myself I would say that there is only one Poisson process with points on a fairly arbitrary set (it's on Wikipedia as Poisson random measure, this term however is very rare in the literature of point processes and random measures). The spatial Poisson process and the one on the real (half-)line are just special cases then. But since Poisson processes on the real line have many particularities and are ubiquitously used in various natural sciences to model random events in time, I think it is a good idea to provide direct access to them and not confuse those readers who are only interested in the one-dimensional version with excessive technicalities. Slartibarti 15:57, 3 October 2007 (UTC)

I've also substantially shortened the intro, moving most of the material into the body of the article. I removed the discussion on memorylessness up through and including the bus analogy as it duplicates material covered in said article. Last, I noticed that this article treated certain processes (e.g. the stars in the sky example) as multi-dimensional Poisson processes that could be more simply characterized by an equivalent probability density function. Using a 1-D case as an example, it can be shown that sampling N events from an i.i.d. uniform distribution on an interval is equivalent to sampling a homogeneous Poisson process on that interval given that there are N points. A similar result holds for more general distributions by modifying the rate function. However these things may be better represented using a spatial probability density function instead of some cooked up rate function for a multi-dimensional Poisson process. Willem 22:23, 7 October 2006 (UTC)

This article seems to have got somewhat confused between Poisson Process and Poisson Distribution. The bombs in areas of London is certainly not a Poisson process (for a start, they were likely to fall at night more than in the day and if a bomb had just arrived there was a good chance of an air-raid being on so a good chance of another soon).

You misunderstand. It's a 2-dimensional spacial Poisson process; there is no time involved! Think of just ONE bombing raid: the number of bombs hitting a particular square mile has a Poisson distribution; the number hitting another, non-overlapping square mile also has a Poisson distribution, and they are independent of each other. The number of bombs falling during any specified time period has nothing at all to do with this. Michael Hardy 20:17, 14 Feb 2005 (UTC)
And now I've read it again, and it seems perfectly clear: it says "two-dimensional" and it talks about non-overlapping plots of ground, so I don't understand how you got so confused. The number of bombs falling during any particular time period was not one of the random variables in this process at all. Michael Hardy 20:20, 14 Feb 2005 (UTC)
I removed the bomb analogy. For the most part, it is a fine analogy, but I thought the rainfall example worked better. After all, the number of bombs is often set a priori, as is sometimes the intended bomb distribution. I also removed the stars analogy for the reasons discussed above relating the uniform distribution to the Poisson process. Willem 22:23, 7 October 2006 (UTC)

It is a classic (and well known) example of a Poisson distribution. I know that classical telephone models are Poisson Processes but the stars example I am not certain of --- surely this is a Poisson distribution again? Have I misunderstood or is there a couple of genuine errors here? If these are errors would anyone mind me rectifying this page?

I think it would also be worth adding the derivation of a Poisson Process as a pure Birth process. Any thoughts?

--Richard Clegg 19:36, 14 Feb 2005 (UTC)

If I may chime in here... It may help to think in terms not of the arrival times, but the interarrival times.
Imagine a spatial Poisson process in one dimension. The distances between the 'bombs' should be exponential, just like the interarrival times of the Poisson process. Instead of making a cumulative count of the number of 'bombs' up to distance x, we just mark down the points at which they dropped, but otherwise the characteristics of this spatial 'process' are just the same. What seems to be the trick here is separating the concept of a stochastic process from the procession in time of a random variable. Think of a stochastic process, instead, as simply a random variable indexed by a (non-random) variable, the latter potentially indicating time, but equally validly, space or some other dimension.
Now think about the pattern of bombings in two dimensions. If the distances between the bombs are described by two-dimensional (independent) exponential distributions (if I recall correctly), we have a two-dimensional Poisson 'process'—it doesn't involve time, but otherwise is Poisson. The number of bombs in a particular area is of course described by a Poisson distribution.
I don't agree that the text as it stands is "perfectly" clear, except perhaps to those with a decent amount of training in the area; no doubt even some of those would disagree. It seems to me that a little more detail would benefit the lay-reader, perhaps by referring back to the introductory comments about Poisson processes in general. Ben Cairns 22:48, 14 Feb 2005 (UTC).
What you are saying (I think) is that the number of bombs N(x,y) in some notional area defined by x,y and an aribtrary origin is a two dimensional Poisson process? I agree. You must admit that this intent is hard to guess from the text. Indeed the text positively argues against that interpretation by saying the number of bombs "on a specified area" which heavily implies to me a constant area. I wouldn't say I was untrained in this area, I just finished teaching a masters class about Poisson processes. I have come across the bombs example and the stars example before as examples of Poisson distributions. Obviously these can be summed to Poisson processes but it is in no way clear from the text that this was what was intended hence my thinking it was the classic confusion of Poisson distribution and process. The text given does nothing to clear up the distinction between a Poisson process and a Poisson distribution and, I would say obfuscates it. However, I don't want to be too pushy about this.
--Richard Clegg 00:45, 15 Feb 2005 (UTC)
I was referring to Michael's comment about it being "perfectly clear", which would only be so if it already made perfect sense to someone before they read it. I did not intend to imply anything about any particular person's training in the area, except that of a 'typical' reader of the Wikipedia. I think that the introductory text does give a definition of a Poisson process that is sufficient to support the bombs example, but it seems too formal for the likely audience. At least, a more intuitive definition could be added to the text for those who just want an idea of what's going on. However, Michael's point is, I think, that the reference to non-overlapping areas and Poisson distributions relates back to the introduction where this stuff is explained. Some of the lack of clarity may be due to this reference, which does no good if a reader is looking at the examples to try and understand the introduction (which, let's say, they largely skipped over). Ben Cairns 02:57, 15 Feb 2005 (UTC).

## Proposed edits

OK. For what it's worth I think the introduction does little to clarify the situation. "The number of arrivals in each interval of time or region in space..." from the word arrival one strongly gets the impression of a fixed space and a change over time which is misleading. The introduction is technically correct but I think it manages to give the conditions for a Poisson Process without actually saying what a Poisson Process is. However, I'm very reluctant to make a change now since the opinion here seems to be against. How about this text very early in the introduction:

A one dimensional Poisson process is a stochastic process defined on some interval. The Poisson process can be thought of as defining the number of occurrences of some event within a subset of this interval. The process A(t) is a Poisson process if the probability of arrivals in some subset of the interval is given by

$P [(A(t+ \tau) - A(t)) = k] = \frac{e^{-\lambda \tau} {(\lambda \tau)}^k}{k!},$

where k is the number of events occurring within [t, t + τ] and λ is a parameter known as the rate parameter. It can be seen that the probability defined is a Poisson distribution with parameter λ τ.

I would also like to slightly edit the examples to at the very least try define "This" in the sentence "This is a two-dimensional Poisson Process"... perhaps to "The number of bombs falling in some area A is Poisson process defined over the area A as it varies. (Though I'm not sure about the last three words, I want to clarify the idea that the area is varying.)

Looking further, this equation is already given on the page for Poisson distribution. Indeed I think the page for Poisson distribution is somewhat clearer to a novice about what a Poisson Process is than this page. However, I am new to wikipedia and reluctant to be too heavy handed. --Richard Clegg 10:21, 15 Feb 2005 (UTC)

I like your text above, although the second sentence is a bit vague and doesn't seem to clarify the distinction between Poisson processes and distributions much more effectively than what is already there. The rest is quite clear. I am coming around to thinking that the whole article could use some rearranging, such as the division of examples into 1-D and multi-D sections, the latter coming after most of the article. Find a place to fit your text in, delete and/or rearrange superceded content, and we'll take it from there. For your suggestion about changes to the bombs example, it seems to me that 'this' needs to be clarified in the rest of the article first, so that the example is clearer, rather than the other way around. Ben Cairns 12:16, 15 Feb 2005 (UTC).
I have made the edit but I am not wholly sure it is clear though I think it does improve clarity, particularly for people not too familiar with the subject. Further editing would be gratefully welcomed. --Richard Clegg 17:10, 15 Feb 2005 (UTC)

## Clarity, examples

There are a few things that are not stated clear enough in the article.

This is what I understand, though I may be wrong in some aspects:

Poisson processes are stateless, that is, the occurrence of past events at particular moments in time has no effect on the chances of new events occurring.

Events are distributed in an uniform manner (assuming constant λ, this should also be explained better in the beginning), but without enforcing uniformity. The uniformity is just a result of randomness. There must be no rule linking different occurrences.

I do not believe stars are distributed by a Poisson law, as they tend to form clusters. This is because they interact with each other by gravitational attraction, which has an effect on the way they are distributed.

For the falling bombs to be distributed according to a Poisson process, they would have to be fired independently. If they were dropped in an air raid, they would fall more or less along a straight line. Now, if the trajectory of the plane and the point of drop are known facts you can still consider this a Poisson process in which the random variable is the deviation caused by the aerodynamics of the bombs and wind, but when the plane trajectory is a random variable, this is no longer a Poisson process.

The rest of the examples I think are OK, but a common characteristic should be pointed out. In general you obtain the Poisson distribution/Poisson process when the events are produced by many independent sources.

I would really like feedback on this, at least to find out if I'm right.

Radu124 11:19, 26 September 2006 (UTC)

==There is an innate lack of knowledge of astronomy in the above, as well as a lack of knowledge of Poisson processes. As for the visible starts in the night sky, with very few exceptions, the appearance of stars clost to one another has NOTHING to do with physical clustering of stars in space. When two stars appear to be close to each other in the sky, almost always they are at drasticly-different radial distances from the Earth and from each other. Hence, their "closeness" is merely an illusion. Let me repeat myself: it had nothing to do with gravitational attractions, physical clustering, etc.

The only known exceptions to this are the Pleiades, which really are a star cluster in our Galaxy, and the two Magellanic Clouds, which are close to our Galaxy, and are also only visible in the Southern Hemisphere, anyway.

Also, when it comes to one-dimensional Poisson processes, a feature of their very nature is that "clusters" of events occur. This follows from the definitions of the Poisson process, and namely from the independence of non-overlapping time intervals. Your mind may rebel against this notion, but this is the way that it is.

Likewise, in two-dimensional (spatial) Poisson processes, the number of events occurring in any non-overlapping areas is independent, and from the very nature, clusters occur. These clusters aren't made by any cause: they just happen, and looking for reasons is a useless exercise.

First of all, although I do not agree with you, thank you for the feedback.
I do not know why you made the assumption that my observation was based on the visible starts in the night sky. I believe that we both should be past that level of education by now. My comment was based mostly on common sense rather than knowledge in astronomy, and my reasoning was that, since the stars interact through gravity, the independence assumption is not met. Of course you don't have to believe me, but you could look for a paper called "Cosmological Distances and Fractal Statistics of Galaxy Distribution". Although not specifically about stars it could prove my point.
Also I didn't do the calculations to prove it, I believe you are also wrong about Poisson Processes. Although clustering may occur on the small scale, averaging over large intervals should decrease the variance. Think about noise reduction in digital photography for example.
By the way, innate means "existing in one from birth; inborn". You didn't expect me to be born with a knowledge in astronomy, did you?
Radu124 13:05, 14 December 2006 (UTC)

sorry, by mistake I have removed something:

Under the "Examples" section, you list "The number of photons hitting a photodetector". This is misleading and produces paradoxes. A better wording would be "The number of times a photodetector discharges when exposed to light." 67.188.93.81 (talk) 19:56, 30 June 2010 (UTC)J. Justice

## Mistake in one of the examples

I think the blue path sample given has zero probability: one of the jumps seems to have twice the height of the others... Is this correct? —Preceding unsigned comment added by 136.152.170.5 (talk) 05:52, 30 May 2008 (UTC)

Good call, you're the first person to notice this in years. Actually, you'll find that sample Poisson process paths have many events that are very close to one another, and the low resolution image made them to appear simultaneously, so there's nothing fundamentally wrong, but for illustrative purposes it would be better to remove this apparent discrepancy. I generated enough samples until I found one that looked nice. FYI, the Matlab code I use to generate it is... Willem (talk) 17:58, 28 April 2009 (UTC)
N=20;
p=cumsum(-log(rand(1,N)));
plot([0 p(floor((2:2*N)/2))],ceil((1:2*N)/2)-1,'LineWidth',2);


Original text: A common assumption in the study of simple queueing systems is that the times-of-arrival of "customers" is a Poisson process, and that the "service time" for each "customer" is exponentially-distributed. This sees application in a wide range of queueing systems, including computer communication networks.

As far as I know "service times" are not exponentially distributed as the exponential distribution is generally inappropriate for modelling process delay times (you could use Erlang or Weibull for this). It's the inter-arrival time that is exponentially distributed (inter-event time of random arrivals).

Is this correct?

Posiebers 04:07, 08 November 2006 (UTC)

I think it often makes sense to model interarrival times as exponentially distributed. It's far less plausible for service times. In some cases it may be true of service times. That wouldn't necessarily be considered a reason not to use exponentially distributed service times in a mere math exercise to assign to students. Michael Hardy 17:13, 14 December 2006 (UTC)

## Criticism from a non-expert

I read this article, & am still unclear just what is a "Poisson process", beyond a vague sense that it is a process somehow based on random distribution -- & that the term can be categorized under a number of different groups with their own labels. As it reads at the moment, it is clearly an article written by experts for other experts.

It would help me immensely to understand the subject if an example were used in the opening paragraph to explain just what this process is. For example, how is "the number of web page requests arriving at a server" a Poisson process? Are we talking about the volume of requests alone? How the number is determined as some function of the number or a quality of their sources? Or how the number of requests is related to the content or visibility of the server? Or is it about the nature of this number of requests to fluctuate in bursts & pauses over time?

I'm sure those who know what a Poisson process actually is are probably laughing their asses off at these questions -- however, there is nothing in this article to keep a reader from asking stupid questions like these, & becoming disappointed (or worse) at the response. -- llywrch 18:24, 21 February 2007 (UTC)

To the comment author just above (Llywrch): the comment seems to be extremely unprofessional from your side. Instead of inspiring or advising a reader (to read more introductory literature or follow the links) you deride him/her. Is Wikipedia being created only for experts? —Preceding unsigned comment added by 88.118.217.167 (talkcontribs) 22:12, 21 February 2007
Wikipedia is not written for experts; it is intended as an introduction. My comments were directed at those who forget that, & may consider responding in an adversarial manner. If I sound adversarial, it is because of my frustration with this article, & the fact it is written beyond the comprehension of a mere college graduate, self-described omnivorous reader & long-time Wikipedian. -- llywrch 17:02, 22 February 2007 (UTC)
I am frustrated that the mathematical articles in Wikipedia presuppose a certain level of training, in apparent disregard for Wikipedia's being a general encyclopædia. For those of you who say "you only need to spend the effort to study the article carefully and understand it," bear in mind this is a Wikipedia article--not a college textbook. In my experience, the authors of college textbooks seem to spend more effort making the content of their works accessible to a "lay" reader than the average Wikipedia math article. And college courses, unlike Wikipedia, have "prerequisites." Reading Wikipedia has no prerequisites; therefore, the articles ought to be written in a manner as to be accessible by any reader with a high-school education. 69.140.164.142 05:32, 27 March 2007 (UTC)

It would probably help if someone included the motivation of the Poisson process as the limit of the binomial distribution, which is an intuitive way to understand many of the properties of the Poisson process (particularly the whole memoryless / stationary thing). GAdam 04:29, 26 April 2007 (UTC)

I would like to add a derivation of the homogenous Poisson process equation ... with some discussion. The derivation isn't that hard actually, it requires only some very basic combinatorics of permutations, and a limit to infinity ... which then requires a variable substitution to produce a limit to zero, and using L'Hopital's rule. It isn't something that a typical high-schooler can do, but it still should be informative. Does anybody object? —Preceding unsigned comment added by LeeHarrison (talkcontribs) 19:08, 1 October 2008 (UTC)

A well-known example is radioactive decay of atoms. Many processes are not exactly Poisson processes, but similar enough that for certain types of analysis they can be regarded as such; e.g., telephone calls arriving at a switchboard (if we assume that their frequency doesn't vary with the time of day), page view requests to a website, rainfall or radioactive decay.

So is it or isn't it? 83.146.15.165 (talk) 01:23, 13 March 2009 (UTC)

## Concerns

I have a concern about the definition: it states that the increments are stationary, but that is true only w.r.t. homogeneous Poisson process, not in general. Sorry if my comment is misplaced. (Oct 29, 2009) —Preceding unsigned comment added by 128.2.90.104 (talk) 19:26, 29 October 2009 (UTC)

Added minor rewording that I hope deals with this point. Melcombe (talk) 10:08, 30 October 2009 (UTC)

Another user: I noticed the same issue. It should be clarified somewhere that the stationary increments property in the definition applies to only the Homogeneous case. — Preceding unsigned comment added by 69.251.251.192 (talk) 20:07, 29 September 2013 (UTC)

## "Discrete-time counterpart"

The text currently states: "The Poisson process is a continuous-time process: its discrete-time counterpart is the Bernoulli process."

But the Poisson process is a counting process whereas the Bernouli Process is not.

Correct me if I'm wrong but I think it would be more accurate to state that "its discrete-time counterpart is the Binomial counting process" and perhaps also add in parentheses for non-experts: (the cumulative sum of a Bernoulli process)

watson (talk) 00:52, 3 August 2010 (UTC)

I see your point. But when I think of a two-dimensional Poisson process, I think of those little blips scattered randomly about the plane. Viewed as a "counting process", the two-dimensional Poisson process assigns a count to every set of positive measure in the plane (and the probability distribution of the count is of course a Poisson distribution). This count is not "cumulative" as I usually think of that term. So I think you're right if you're talking about a one-dimensional case, but definitely your parenthetical remark should be included for clarity. Michael Hardy (talk) 03:08, 3 August 2010 (UTC)

## Non-Homogeneous Poisson Process

I am not an expert, but reading scientific articles using non-homogeneous poisson processes I often see an alternative definition: instead of using an integral over lambda to the power of k, they use a product over all events:

$P [t_1, t_2, ..., t_k] = \frac{e^{-\lambda_{a,b}} \prod_{j=0}^{k} \lambda(t_j) }{k!}$

Here of course the time the events occured has to be known. Maybe someone knowledgable could integrate this formulation into the article? —Preceding unsigned comment added by Moritz.blume (talkcontribs) 08:34, 17 September 2010 (UTC)

## Generating time intervals

I think the article should also contain a generator which defines a random variable distributed like the interval between two events from a uniform random variable. Similar to the way the Box–Muller_transform generates normal distributed variables. I think the formular looks like this, but you need to verify it/find a source: t=-tau*ln(1-u) --- W —Preceding unsigned comment added by 84.151.207.53 (talk) 21:25, 28 October 2010 (UTC)

## Possible Error in Definition

The definition starts out by stating four properties viz. N(0)=0, independent increments, stationary increments and no simultaneous occurrences. I seriously doubt if the Poisson distribution on N(t) and the exponential waiting time are consequences of this definition. A better definition in my view should say the following:

A counting process N(t) is called a Poisson process when the following conditions hold:

(1) $N(0) = 0$

(2) For any times $0 \leq t_1 < t_2 \leq \infty$, $N(t_2)-N(t_1)$ has a Poisson distribution with parameter proportional to the time difference $t_2 - t_1$.

(3) Given any collection of disjoint time intervals $(t_1, t_2), (t_3, t_4), \cdots, (t_{k-1},t_k)$, the Poisson counting increments $N(t_2)-N(t_1), N(t_4)-N(t_3), \ldots, N(t_k)-N(t_{k-1})$ are independent random variables.

Now, as a consequence of this definition, one can actually show that the interarrival times are exponentially distributed and location of occurrences on any fixed interval is uniformly distributed on that interval. (See, for instance, Sidney Resnick, Adventures in Stochastic Processes, Birkhauser.) Ingle.atul (talk) 06:13, 2 May 2012 (UTC)

The position seems to be that there are several different ways of characterizing a Poisson process that, with some additional mathematics, can be shown to be defining essentially the same stochastic process. One that is closer to that in the article but not identical is given by Cox & Lewis (1966) The Statistical Analysis of Series of Events, Methuen (Section 2.1). Clearly your proposal relies on a pre-existing definition of the Poisson distribution, which at least some of the others do not (Cox & Lewis's does not). There is also the question of starting the article with things that are understandable by a general reader, per WP:MOSMATH, where your proposal (as it stands) relies on a maths-symbol-heavy approach. It is unfortunate that previous editors have not included a source for the basic definition in the article. The question of whether the specific thing "Poisson distribution" appears in the definition of the Poisson process is important in the context of Wikipedia because the Poisson distribution article uses the example of its occurence (as on outcome of, not defining) in the Poisson preocess as a reason why the distribution is important, so there is a danger of the these arguments becoming circular. Melcombe (talk) 07:50, 2 May 2012 (UTC)
In think I see your point. The current definition is perhaps paraphrasing an alternative (and mathematically precise) definition of the Poisson process that says the following: In a small time interval [t, t+h) the probability of seeing 1 hit is a constant times h plus some negligible higher order terms in h. Also, the probability of seeing zero hits is one minus that. And the probability of seeing more than 1 hit decays to zero faster than a linear function of h.

Ingle.atul (talk) 16:59, 4 May 2012 (UTC)

## Characterisation via Orderliness and Memorylessness

In the Characterisation section, there was a proof and a comment pointing out the flaws of the proof. I agree with the comment that the proof does not seem correct, but since that doesn't belong in the article, I removed both.

If anyone knows how to fix, this please go ahead. It seems that the original author of the proof meant the density instead of the distribution. However I don't understand the step in line 3, since there was no mention of lamba beforehand. Similarly I don't understand why the author could replace $Pr[N_x=0]$ with $\exp(-\lambda x)$.

I have reproduced proof and comment below.

Proof : Let $\tau_1$ be the first arrival time of the Poisson process. Its distribution satisfies

\begin{align} Pr[\tau_1=x]=&\lim_{dt\to 0}\frac{Pr[N_{x+dt}>0,N_x=0]}{dt}\\ =&\lim_{dt\to 0}\frac{1-Pr[N_{dt}=0]}{dt}Pr[N_x=0]\\ =&\lim_{dt\to 0}\frac{1-(1-\lambda dt +O(dt^2))}{dt}\exp(-\lambda x)\\ =&\lambda\exp(-\lambda x) \end{align}

Note that $Pr[\tau_1 = x]$ must be equal to zero for any $x$, since this is always the case for continuous random variables. Thus, the reasoning above is wrong! For instance, the probability that $\tau_1 = 0$ is equal to $\lambda$ which could be greater than one! This raises the question, of course, why such a result would be presented at all, instead of sound and correct statements. — Preceding unsigned comment added by 87.77.25.252 (talk) 09:30, 17 October 2012 (UTC)