Talk:Poisson distribution

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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)

Contents

[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)

[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)

[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. —The preceding unsigned comment was 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)

[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)

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