Talk:Poisson distribution

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[edit] Misprint

I think there is a misprint in the 'Entropy' section of the right panel, in the formula ( a png-file) that gives a large lambda asymptotic for the entropy: Logarithm symbol should be 'ln' rather than 'log' . Otherwise, expressions valid for any lambda and its limiting case look contradicting each other. The present notation create confusion since definitions of entropy may use different bases of logarithm.


Torogran (talk) 10:13, 19 March 2008 (UTC)

[edit] Error?

I'm pretty sure there's an error in this article. Einstein demonstrated the existence of photons while investigating the photoelectric effect, not blackbody radiation. Planck had already dealt with blackbody radiation a few years earlier.

If you know that to be a fact, go ahead and change it. (Such a fact isn't really essential to the topic of this article.) Michael Hardy 21:19, 6 December 2006 (UTC)

Is the number of errors in Wikipedia page really Poisson distributed? With certain assumptions of the process producing the errors, it might be for pages of the same length, but hardly for all pages. Or maybe this was a troll? —Preceding unsigned comment added by 193.142.125.1 (talk) 19:29, 26 May 2008 (UTC)

[edit] Lamda

Strangely, λ doesn't display as \lambda\, on my computer and I don't have a clue what the \, is for.

Also, I moved the normal distribution approx. into the connections to other dist. section to be consistent with the binomial distribution.

Frobnitzem 21:04, 7 September 2006 (UTC)

The \, causes it to render properly on some browsers. Michael Hardy 21:06, 5 February 2007 (UTC)

On a very unrelated note, it seems as if The Economist has taken the graphics for the Poisson/Erlang/Power law/Gaussian distributions from Wikipedia and published them in an article: Article: [1] and image: [2]

The limit of the binomial distribution isn't so much how the Poisson distribution arises as one example of a physical situation that the Poisson distribution can model fairly well. It far more often arises as the limit of a wide number of independent processes, which can in turn be modelled by the binomial distribution - but the model isn't the thing.

As it happens, it's a lot more illuminating and a better look at the causality to examine this limit of a wide number of independent processes using differential equations and generating functions, but it's simpler to use the binomial distribution approach. PML.

The comment above definitely could bear elaboration! Michael Hardy 01:45 Feb 5, 2003 (UTC)

Well, for instance consider how many breaks a power line of length l might have after a storm. Suppose there is an independent probability lambda delta l of a break in any stretch of length delta l. (We know this is crawling with assumptions; if we do this right - like the better sort of economist - in any real case we will check the theory back to outcomes to see if it was really like that in the first place.)

Anyhow, we pretend we already have a general formula and put it in the form of a Probability generating function P(lambda, l, x). Then we get an expression for P(lambda, l + delta l , x) in terms of P(lambda, l, x) and P(lambda, delta l , x). When we take the limit of this we get a differential equation which we can solve to get the Poisson distribution.

If people already know the slightly more advanced concept of a Cumulant generating function we can rearrange the problem in that form, and then the result almost jumps out at you without needing to solve anything (a Cumulant generating function is what you get when you take the logarithm of a probability generating function).

Actually, the cumulant-generating function is the logarithm of the moment-generating function. Michael Hardy 22:05, 2 Apr 2004 (UTC)

I have heard that the empirical data that was first used for this formula was the annual number of deaths of German soldiers from horse kicks in the 19th century. PML.

  • I'm not sure that this isn't just the same as what is on the page, just with different maths. I disagree with PML (but am open to being convinced otherwise) and think the binomial is a great place to start a derivation of the Poisson distribution from. It is exactly the appropriate approximation for nuclear decay, phones rining, et cetera. I would also use it for the above example. --Pdbailey 13:21, 31 Aug 2004 (UTC)

Concerning the source of the horse-kick data, see Ladislaus Bortkiewicz; it was his book The Law of Small Numbers that made that data-set famous. 131.183.84.78 02:25 Feb 5, 2003 (UTC)

I've seen this approach via differential equations before, but I don't think it's a reason not to include the limit theorem. For that matter, I still think an account of the limit theorem should appear earlier in the article than anything about differential equations or cumulant-generating functions. Michael Hardy 02:31 Feb 5, 2003 (UTC)

The word "arise" really only tells us that we can do the algebra this way, not that the process is itself like this.

My concern was that the wording suggests that it all somehow comes out of the Binomial distribution, when that is simply yet another thing that can describe/model the same sort of underlying processes. You would expect the limit of the binomial distribution to work, but only because it is itself modelling the same processes; but it only does that when you plug the right things in, i.e. taking the limit while you keep the expected values where you want them. You can have a binomial distribution that converges to other limits under other constraints. PML.

None of which looks to me like a reason why the limit theorem should not be given prominence before cumulants or differential equations are mentioned. I agree that the "constraints" do need to be emphasized. Michael Hardy 02:41 Feb 5, 2003 (UTC)

I think you're missing my point. I'm not saying you shouldn't mention these things early on. Only, you shouldn't make them look like where the Poisson distribution comes from, the underlying mechanism. You could easily use these things to show how to calculate it, to get to the algebraic formula, while stating that these are merely applying underlying things which will be bought out later. It's the word "arise" in the subtopic introduction I'm uncomfortable with, not what you're doing after that.

An analogy: it's a lot easier to state a formula for Fibonacci numbers, and prove that the formula works with mathematical induction, than to derive it in the first place - and it was probably derived in the first place by using generating functions. So you introduce the subject with the easy bit but you don't make it look like where you're coming from. PML.

I don't know the history, but to me it is plausible that the limit theorem I stated on this page is how the distribution was first discovered. And if you talk about phone calls arriving at a switchboard, it's not so implausible to think of each second that passes as having many opportunities for a phone call to arrive and few opportunities actually realized, so that limit theorem does seem to describe the mechanism. Michael Hardy 17:20 Feb 5, 2003 (UTC)

I am a dunce, but wouldn't the number of mutations in a given stretch of DNA be a binomial distribution, since you have discrete units? You couldn't very well have a nice Poisson process with a DNA stretch of only 4 base pairs... on the other hand maybe I don't know what I'm talking about... Graft 21:14, 2 Apr 2004 (UTC)

It would be well-approximated by a Poisson distribution if the number of "discrete units" is large, and using a Poisson distribution is simpler. Michael Hardy 21:23, 2 Apr 2004 (UTC)

I've been developing a new distribution curve to describe the number of correctly ordered random events when the order of each event is relative to the other events. In other words, 'A' comes before 'B', but there may be any distance between 'A' and 'B'. The pattern also demonstrates that when given a portion of the relative sequence, the probability of getting the unknown portion correct increases by an amount dependent upon the distance between the given events. In fact, given only one relative order, you have a better chance at getting the rest of the sequence correct, when the known relative order includes the endpoints of the sequence. The least valuable given would be consecutive events. I believe that this distribution curve will have value when analyzing DNA sequences. I've also determined that the Binomial Distribution is not appropriate for assessing ordered events (i.e. Grading a student's list of presidents in historical order). If anyone is interested, I am willing to discuss my work and provide my argument against use of the Binomial Distribution to compare the homology of DNA sequences. You may contact me through johnnleach@hotmail.com, and begin the title with "Rhonda give this to John". My wife has taken over my email account. After establishing contact, I can give you a better means of contacting me. User: JNLII, May 8, 2008. —Preceding unsigned comment added by JNLII (talk • contribs) 16:27, 8 May 2008 (UTC)

[edit] Waiting time to next event.

In the waiting time to the next event

P(T>t)=P(N_t=0)=e^{-\lambda t}.\,

This looks like it isn't normarmalized. since there should be a λ out in front. Am I wrong? Pdbailey 03:47, 11 Jan 2005 (UTC)

Yes; you're wrong. The normalizing constant should appear in the probability density function, but not in this expression, which is 1 minus the cumulative distribution function. Michael Hardy 03:50, 11 Jan 2005 (UTC)

[edit] Parameter estimation

I'm confused about the recent edits to the MLE section. I'm under the distinct impression that the sample mean is the minimum-variance unbiased estimator for λ, but a combination of ignorance and laziness prevents me from investigating this myself. Could someone please enlighten me? --MarkSweep 07:07, 15 May 2005 (UTC)

Evidently when I wrote it, I was also confused. I think its right this time, please check the derivation. I didn't put in the part about "minimum variance" because I can't prove it quickly, and I haven't got a source that says that, but it would be a good thing to add. PAR 14:07, 15 May 2005 (UTC)
This MLE is unbiased, and is the MVUE. MLEs generally are often biased. Michael Hardy 22:42, 15 May 2005 (UTC)

[edit] Poisson Distribution for Crime Analysis?

Is a Poisson distribution the best one for describing the frequency of crime? Before I add it as an example on the main page, I’d like to post this for discussion.

Recently, I've been trying to use the normal distribution to approximate the monthly statistics of the eight "Part I" crimes in the ten police districts of San Francisco. But the normal distribution is continuous and not discrete like the Poisson. It also doesn't seem appropriate for situations where the value of a crime like homicide is zero for several weeks.

My goal is to approximate the occurrences of crime with the appropriate distribution, and then use this distribution to determine whether a change in crime from one week to the next is statistically significant or not.

Distinguishing between significant change and predicable variations might help deploy police resources more effectively. Knowing the mean and standard deviation of the historical crime data, I can compare a new week’s data to the mean, and - given the correct distribution - assess the significance of any change that has occurred. But is the Poisson distribution the one to use?

Also, how do I take into account trends? Does the Poisson distribution assume that the underlying process does not change? This may be a problem because crime has been going down for years.

- Tom Feledy

Well, IANAS, but my advice would be to first set up a simple Poisson model and assess its goodness of fit. My guess is there could easily be several problems with a simple Poisson model: First of all, it has only a single parameter, so you cannot adjust the mean independently of the variance; you may want to look into a Poisson mixture like the negative binomial distribution as an alternative with more parameters. Second, as you point out yourself, zero counts (fortunately) dominate for many types of crimes. This suggests that you need a zero-inflated or "adjusted" distribution, like a zero-inflated Poisson model in the simplest case. Finally, if you have independent variables that could potentially explain differences in the frequency of certain crimes, then a conditional model (e.g. Poisson regression analysis) will be more appropriate than a model that ignores background information and trends. --MarkSweep 02:26, 31 May 2005 (UTC)
You might also look at a non-constant rate parameter. But estimating that might be delicate. Michael Hardy 02:52, 31 May 2005 (UTC)

[edit] Two-argument gamma function?

The article as it stands uses a two-argument function called Γ to define the CDF. The only gamma function Wikipedia knows about takes only one argument. What is this two-argument function? Thanks! — ciphergoth

It's the incomplete Gamma function. The Poisson CDF can be expressed as
\Pr[X\leq k] = Q(k+1, \lambda) = \frac{\Gamma(k+1,\lambda)}{k!} \!
where Q is the upper regularized Gamma function and Γ is the upper incomplete Gamma function. Given that
\Gamma(1,\lambda) = \exp(-\lambda)\!
and
\Gamma(k+1,\lambda) = k\,\Gamma(k,\lambda) + \lambda^k \exp(-\lambda)\!
one can easily show by induction that
\sum_{j=0}^k \Pr[X=j] = \frac{\Gamma(k+1,\lambda)}{k!}\!
holds. --MarkSweep✍ 16:30, 14 October 2005 (UTC)

Hah! I had exactly the same question. It took me ages to find the answer - via the Wolfram Mathematica website among others - so I've updated the page at that point. I hope consensus is it goes well there. [User: count_ludwig (not yet registered)] 18:30, 17 July 2007 (UTC)

I am still doubting the accuracy of this CDF. I tried in Matlab and it is actually the lower incomplete function which gives the same values as the built-in CDF. Moreover, I agree with 65.96.177.255, by looking at the bounds of the integral, the lower incomplete function makes more sense than the upper one. Could MarkSweep provide the complete proof? Nicogla 11:46, 21 September 2007 (UTC)

[edit] X~Poisson(λ)

When I was studying statistics (few years back now), the notation used in the independent references we worked from identified the distribution as Po(λ) rather than Poisson(λ). Of course, if someone disagrees, feel free to put it back as it was. Chris talk back 01:58, 31 October 2005 (UTC)

Actually, I do disagree. To a certain extent it's an arbitrary decision, but consider the following factors: (1) I think neither "Po" nor "Poisson" is an established convention, so there is no reason to prefer one over the other; (2) "Poisson" is more descriptive and less confusing; (3) "Poisson" is what we use in a number of other articles (e.g. negative binomial distribution). I'd say there are no reasons to prefer "Po", at least one good reason to prefer "Poisson", plus a not-so-good reason (inertia) to stick with "Poisson". --MarkSweep (call me collect) 04:59, 31 October 2005 (UTC)
When I've seen it abbreviated, I think I've usually seen "X ~ Poi(λ)", with three letters. I'm not militant about it, but I prefer writing out the whole thing. Michael Hardy 22:20, 31 October 2005 (UTC)
Whatever. Personally I think Poi just doesn't look right, but that's a matter of opinion. Chris talk back 23:29, 1 November 2005 (UTC)

[edit] Erlang Distribution

There's a refrence to erlang distribution, but the article does not mention the mutual dependence between Erlang Distribution and Poission Distribution. That is, the number of occurrences within a given interval follows a Poission distribution iff the time between occurrences follows an exponential distribution. (unsigned by user:Oobyduby)


[edit] CDF is defined for all reals

It has to be a piecewise constant function with jumps at integers. —The preceding unsigned comment was added by PBH (talk • contribs) .

I don't see why. Most books I have referenced (Casella and Berger's Statisitical Inference, for example) give the range as non-negative integers. Why should it be piecewise constant? --TeaDrinker 16:12, 30 May 2006 (UTC) Ah, looking at the graph again I see the error. Indeed the CDF should be piecewise constant, not interpolated as has been done. My mistake. --TeaDrinker 16:15, 30 May 2006 (UTC)
How does this look?
It does not quite look like the other (pdf) plot. However it does do the stepwise progression. Cheers, --TeaDrinker 16:32, 30 May 2006 (UTC)
I would do away with the vertical pieces. If you do it in MATLAB, you could probably use something like plot( x, y, '.' ); At any case, this is much better, at least mathematically if not aesthetically. PBH 16:56, 30 May 2006 (UTC)

To me, the mass function seems far easier to grasp intuitively than the cdf, so I wouldn't mind if no cdf graph appeared. In the mean time, I've commented out the incorrect one that appeared. Michael Hardy 02:02, 31 May 2006 (UTC)

I've posted a CDF and then removed one that was grossly misleading. The problem with the pdf and cdf here is that it isn't clear that the lines are eye guides and do not represent actual mass. This error is more problematic in the case of the CDF because there is no reason for the eye guide, the cdf (unlike the pdf) has support on the positive real line. The plot I posted also has problems. there should be no vertical lines, and there should be open circles on the right edges of each horizontal line and closed circles at the left edge. Pdbailey 00:17, 2 June 2006 (UTC)

okay, I added these features. If you want to post one that you think looks prettier, please be sure that it meets the definition of the CDF. Pdbailey 02:46, 2 June 2006 (UTC)

[edit] Parameter estimation

In the parameter estimation section it is surely not necessary to appeal to the characteristic function?

Expectation is a linear operator and the expectation of each k_i is lambda. Therefore the sum of the expectations of N of them chosen randomly is N lambda and the 1/N factor gives our answer. Surely the characteristic function here is needless obfuscation? --Richard Clegg 14:49, 14 September 2006 (UTC)

I've fixed that. It was very very silly at best. Someone actually wrote that if something is an unbiased estimator, it is efficient and achieves the Cramer-Rao lower bound. Not only is it trivially easy to give examples of unbiased estimators that come nowhere near the CR lower bound, but one always does so when doing routine applications of the Rao-Blackwell theorem. Michael Hardy 20:44, 14 September 2006 (UTC)

[edit] Graphs

the poisson graphs dont look right. shouldnt the mean be lamba? it doesnt look like it from the graphs if so. —Preceding unsigned comment added by 160.39.211.34 (talk • contribs)

Well, it's quite hard to visually tell the mean from a function plot, but fortunately in this case the mode is also floor(λ), and in the case of λ an integer there is a second mode at λ−1. I don't see anything that's visually off in Image:Poisson distribution PMF.png. --MarkSweep (call me collect) 07:58, 5 December 2006 (UTC)

[edit] Poisson model question

Does a material requisiton filling process fit into a poisson model? A wrong requisition is generated hardly ever, so p is very small. X= "Requisitions with errors" —The preceding unsigned comment was added by 200.47.113.40 (talk) 12:40, 19 December 2006 (UTC).

[edit] Poisson median formula source and correctness

Implementing the Poisson distribution in C++, I find that the quantile(1/2) does not agree with the formula given for the median. The media is about 1 greater than the quantile(half). Is this formula correct? What is its provenance. Other suggestions? Thanks

Paul A Bristow 16:52, 19 December 2006 (UTC) Paul A. Bristow

Have you tried with the GSL (GNU Scientific library): [3] and [4]? --Denis Arnaud (talk with me) 18:36, 22 March 2007 (UTC+1)
I have checked it by numerical calculation via a self-written program and using formulae from the Numerical Recipes. The formula in the table is almost correct, but 0.2 has to be replaced with 0.02. Then it is fairly correct (the absolute error is less than about 0.001, the relative one even smaller).--SiriusB 14:06, 13 June 2007 (UTC)

[edit] UPPER incomplete gamma funct?

Doesn't it make sense that the cdf would be the lower incomplete gamma function rather than the upper? Am I missing something?

65.96.177.255 23:27, 4 February 2007 (UTC)blinka

[edit] mode

Isn't the mode both the floor and if lambda is an integer, the next lower integer as well? Pdbailey 22:23, 26 March 2007 (UTC)

I've add this several times and it has been deleted without comment in the edit summary, please post here if you disagree! Pdbailey (talk) 02:48, 12 February 2008 (UTC)
By the way, if that is the case, then another way to write it is simply \lceil{\lambda-1}\rceil. Chutzpan (talk) 16:32, 23 March 2008 (UTC)
Chutzpan, I can see what you are saying and it is notationally smaller when typeset, but this property is mainly cherished over clarity by mathematicians. I think it is clearer to point out that the distribution is only bimodal when lambda is an integer since otherwise the reader has to take a minute to figure that out. Pdbailey (talk) 17:10, 23 March 2008 (UTC)

[edit] Einstein

"Albert Einstein used Poisson noise to show that matter was composed of discrete atoms and to estimate Avogadro's number; he also used Poisson noise in treating blackbody radiation to demonstrate that electromagnetic radiation was composed of discrete photons."

These claims need their respective citations. They are far from being "common knowledge" about Einstein, at least in the specific wording that the claims use. I am removing them until the proper citations are given.

Even with citations, this is too specific for this article. Many thousands of scientific endeavors use Poisson processes of one kind or another. McKay 04:03, 12 April 2007 (UTC)
I think that, if the citations support the claims, the claims are historically interesting. However, I'm not sure if the claims are fully supported. For example, did Einstein's 1905 Brownian motion article talk about a Poisson noise or rather about a Gaussian noise? Was the editor referring to this or to another article? And with regard to the claim about the blackbody radiation, the first entry in this talk page had already doubted about its validity. I will ask editors of Wikipedia's Albert Einstein article anyway. (Sorry, I forgot to sign last time. Another Wikipedian 05:36, 12 April 2007 (UTC))

[edit] Formula in complex analysis

I know very little about statics, but it seems to me the article does not discuss the poisson formula in complex analysis, which I know. I am thinking of renaming a newly created Schwarz formula to poisson formula replacing the redirect. Any feedback? -- Taku 09:57, 28 April 2007 (UTC)

[edit] section order

I propose that the the first section after the introduction regarding shot noise should be folded into the examples section as a bullet. It is already covered in the article on shot noise very throughly and I'm not sure what's so much more interesting about this example than any of the others. Pdbailey 13:59, 17 May 2007 (UTC)

[edit] Posterior of a Gamma-distributed prior for a poisson parameter

Somehow, the formula given for the posterior distribution makes little sense (in the sense that it is not stable - it will not yield agreeing distributions if fed with data specifically tailored to meet the prior). I'd have to check with my textbooks (I will when time allows), but it seems to me that there's a typo - browsing the net, I found a similar (yet different) one:

\lambda\approx\text{Gamma}\left(\alpha+\sum_{i=1}^nk_i,\frac{1}{\frac{1}{\beta}+n}\right)

That came from here, if you're interested (and I don't really know how trustworthy that page is... so I'd still check some textbooks). In any case, that new formula clearly fits better with the assertion that

The posterior mean E[λ] approaches the maximum likelihood estimate \widehat{\lambda}_\mathrm{MLE} in the limit as \alpha\to 0,\ \beta\to 0.

Since

\widehat\lambda_\mathrm{MLE}=\frac{1}{n}\sum_{i=1}^nk_i=E\left[\text{Gamma}\left(\sum_{i=1}^nk_i,\frac{1}{n}\right)\right]

(when n\to \infty, which is another change)

Correct posterior is \lambda \sim \mathrm{Gamma}(\alpha + \sum_{i=1}^n k_i, \beta + n). \! Note that Gamma is usually parameterised by shape and rate (inverse scale) when used as a conjugate prior for the Poisson, not by shape and scale. —Preceding unsigned comment added by 222.152.28.123 (talk) 07:18, 30 May 2008 (UTC)

[edit] Web server example: Repeat visitors vs. first-time visitors

The "Occurrence" section currently reads:

Examples of events that may be modelled as a Poisson distribution include: ...
• The number of times a web server is accessed per minute.

Since website visitors tend to click around a multi-page website at a click-rate which differs from the arrival rate, may I suggest any of the following amendments:

  1. The number of times a web page is accessed per minute.
  2. The number of times a web server is accessed per minute by new, unique visitors.

-- JEBrown87544 04:18, 2 September 2007 (UTC)

[edit] Maximum of the distribution

It would be useful to the information on calculating the maximum of the distribution.

The Poisson distribution has the maximum between (λ − 1) and lambda, because poisson(x, lambda) = poisson(x + 1, lambda) gives the result x = lambda - 1. We look for two equal probability values that are distant from one another by 1. This give as a pretty good clue where the maximum is.

Since lambda doesn't have to be an integer, we have to consider consider floor(lambda - 1), and ceil(lambda) as the possible values for the maximum. Also (floor(lambda - 1) + 1) can be the maximum, so we consider this too. It seems safe to assume that there are three values for the maximum to consier:

floor(lambda - 1) floor(lambda - 1) + 1 floor(lambda - 1) + 2

But we need to make sure that (λ − 1) is not negative.

The C code for calculating the maximum is:

int poisson_max(double lambda) {

 assert(lambda > 0);

 int k_ini = int(lambda - 1);
 if (k_ini < 0)
   k_ini = 0;

 int k_max = k_ini;
 double f_max = gsl_ran_poisson_pdf(k_max, lambda);

 // We choose the max of k_ini, (k_ini + 1), and (k_ini + 2).
 for(int k = k_ini + 1; k <= k_ini + 2; ++k)
   {
     double f = gsl_ran_poisson_pdf(k, lambda);
     
     if (f > f_max) 
       {
         f_max = f;
         k_max = k;
       }
   }
 
 return k_max;

}


Thanks & best, 83.30.152.116 19:16, 6 October 2007 (UTC)Irek

This is called the mode and it is on the page as such. It is lambda or lambda and the number one less if it is an integer. Pdbailey 20:39, 6 October 2007 (UTC)

Thanks for the info! —Preceding unsigned comment added by 83.30.152.116 (talk) 21:01, 6 October 2007 (UTC)

sorry, should be floor lambda or lambda and lambda + 1 in the case of an integer. Pdbailey 21:30, 6 October 2007 (UTC)


[edit] Pronunciation

I think the article should mention pronunciation of poisson. —Preceding unsigned comment added by 212.120.248.128 (talk) 22:04, 9 December 2007 (UTC)

[edit] technical?

This article may be too technical for a general audience.
Please help improve this article by providing more context and better explanations of technical details to make it more accessible, without removing technical details.

The introduction to this article is excellent, but given the importance of the Poisson distribution to many fields such as, for example, call-center management (where the average practitioner may not necessarily have a mathematical background), it would be desirable to make the rest of the article more comprehensible. 69.140.159.215 (talk) 12:52, 12 January 2008 (UTC)


May I suggest an modification to the graphs, one that presents it in terms of concrete things rather than abstract symbols (k and lambeda) that need to be looked up?

Imagine that this is being read by someone who isn't even familiar with the convention of expressing probabilities as a proportion of 1. Only one axis is labeled. One page further down, it is explained that "k is the number of occurrences of an event". What? When? Where?

If we need to have k and lambeda in there, let's have axis labels and a legend that define them:


lambeda = number of events we expect to observe (on average)

k = number of events we really observe

p = probability of observing k events


We could have a caption something like

"Example of use: If one event is expected in the observations (e.g. the event happens on average once every decade, and your observations cover a decade) then the chance of having no events in the observations is 0.37 (37%), the probability of seeing one event is 0.37, the probability of seeing two events is 0.18, etc.."

For the second graph, the same, only let it read "one or zero events (k<=1),... two events or fewer... three events or fewer..."

This implies all that the current caption says. By character count it's twice as long, though. Any other suggestions?

If people get the basic idea upfront, from the graphs, they will get much more out of the article.

HLHJ (talk) 19:04, 20 May 2008 (UTC)

[edit] Mode

Why mode is not stated as \lceil\lambda - 1\rceil - it simpler than currently stated formula and as far as I understend equivalent? Uzytkownik (talk) 12:04, 26 March 2008 (UTC)

I've answered this question in the first section on the talk page that is titled, "mode." Pdbailey (talk) 01:58, 27 March 2008 (UTC)

[edit] Time

The definition given here seems very time-centric:

In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

Are there other dimensions besides time and space that could apply? If not I suggest simply saying 'time or space' (volume, area and distance all being spatial). In any case, surely a broader definition should be given. Richard001 (talk) 09:27, 3 April 2008 (UTC)

Richard001, it's a little harder with space because the event usually already occurred. i.e. the number of stars within a certain portion of the sky. Pdbailey (talk) 03:09, 10 May 2008 (UTC)

[edit] Examples

Many of the examples given in the "Occurence" section are probably not Poisson. It might be better to have a much shorter list of easily defensible examples. OliAtlason (talk) 23:02, 18 April 2008 (UTC)

Agreed, be bold! Pdbailey (talk) 03:07, 10 May 2008 (UTC)

[edit] Schwarz formula

Why is Schwarz formula in the See Also section ? How is it relevant to the poisson distribution ? Is someone confusing poisson kernel with poisson distribution ? —Preceding unsigned comment added by 24.37.24.39 (talk) 04:09, 14 May 2008 (UTC)

[edit] The law of rare events

I tried to clarify the meaning of "rare" in this term -- it applies to the (very many) individual Bernoulli variables being summed, not the result. The later "law of small numbers should probably be moved into this section by the way, but I'm not sure how to do it cleanly. Quietbritishjim (talk) 15:32, 9 June 2008 (UTC)