Talk:Poisson process

message: 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)
replies: text: 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  Find a place to fit your text in, delete and/or rearrange superceded content, and we'll take it from there.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)
replies: text:  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)
thread_title: Proposed edits

message: 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)
replies: text: 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)

text: 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)
thread_title:  Mistake in one of the examples 

message: :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)
thread_title:  lead para contradiction 

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

message: 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)
replies: text: 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)
thread_title:  "Discrete-time counterpart" 

message: 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 (talk • contribs) 08:34, 17 September 2010 (UTC)
thread_title: Non-Homogeneous Poisson Process

message: 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)
thread_title: Generating time intervals

message: 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)
replies: text: 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)
replies: text: 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)
thread_title: Possible Error in Definition

message: 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)
thread_title: Characterisation via Orderliness and Memorylessness

message: In Poisson_process#Properties, is the "citation needed" really sound, as there are two references to the Markov processes, which contain the explanation of the statement (relationship between Poisson, memoryless, random and Markov). I suggest to remove the request for a citation there. DirkHoffmann (talk) 18:29, 1 March 2015 (UTC)
thread_title:  Citation needed?