Talk:Poisson distribution/Archive 2

Degree of spread measured in... what units?
Sorry, I'm still struggling with understanding the concept, and I'd hoped for the description to be more precise. Compare this to other things that measure spread, for example, STD or SE, they use the same units as the variable being measured. I.e. if you measure hight of human population in centimetres, then the standard error will measure the spread of values in your measurements in centimetres as well. Poisson distribution doesn't seem to use the same units... or does it? I'm struggling to makes sense of the numbers I'm getting from a formula, and it just doesn't add up to anything :( sorry. 109.65.144.222 (talk) 19:14, 25 May 2014 (UTC)
 * Poisson distribution, in contrast to the normal (and many other) distributions, is for integers only. Thus the notion of units is not applicable; "someone typically gets 4 pieces of mail per day" — could you say this in different units? meters? kilograms? No, never. "Piece" is not a unit. In physics (and chemistry, etc) such quantities are treated as dimensionless. They are the same in all systems of units (SI, CGSE etc). Boris Tsirelson (talk) 21:03, 25 May 2014 (UTC)
 * Well, you misunderstood me. I don't mean it has to be some special unit (as in physics), it has to relate to the units of the feature being researched. The example you quote: someone gets 4 pieces of mail - then the the spread could be measured in: a) pieces of mail. b) ratio of how many pieces of mail are being predicted to some etalon ratio, such as maximum entropy for example. These are two explanations that I've been pondering, but as I've said - none of them makes sense, if you plug in the numbers. To be more concrete, the example I stole from http://stattrek.com/probability-distributions/poisson.aspx it goes like this: *The average number of homes sold by the Acme Realty company is 2 homes per day. What is the probability that exactly 3 homes will be sold tomorrow?*. And plugging the numbers into the formula I get 0.18045. All hunky-dory, but I don't know what does this number mean when applied to spread. When interpreted as frequency or probability - its units are "chances" or "units of frequency", so intuitively I can interpret it as "if I were to sell property, then approximately one time in six I'd sell three items whereas on average I'd sell two" or something to that effect. But when it measures spread - shouldn't it reflect on how the error scales with the measurement? I can't tell from looking at 0.18045 what is the error, and whether the data is widely scattered or is dense because I'm lacking the ability to interpret this number in any such way. Hope I make it clear! 109.65.144.222 (talk) 22:19, 25 May 2014 (UTC)
 * Not very clear to me. It seems, you really do not want a distribution (that is, collection of probabilities) but rather STD. Then, what is the problem? STD for Poisson is the square root of lambda. Boris Tsirelson (talk) 05:44, 26 May 2014 (UTC)
 * Or do you want to say that, in your opinion, the phrase "it predicts the degree of spread around a known average rate of occurrence" is misleading? Then just say so. Boris Tsirelson (talk) 05:48, 26 May 2014 (UTC)
 * Yes, I see now. I'll try a different approach. STD is a measure of spread. Let's assume the "worst" case of the normal curve where the "shoulders" are on the same level as the peak. I.e. the curve degenerated to a straight line (with STD approaching infinity). This gives me the worst predictive power possible in this setting. The more I progress towards "lower shoulders" and "narrower peak", the better I get at predicting the outcomes. Thus I can compare spreads: the one with lower shoulders and narrower peak is "more dense" - it gives me better predictive power. Now take my previous example with Acme Realty. Suppose now I'm selling on average 5 pieces of property and want to predict selling exactly 6. This gives me the chance of 0.14622. What I don't know is am I now better (more precise) at predicting the outcome or not? And if I can tell, then how do I know?
 * Oh, I just saw your other comment. Well, kind of. I don't really understand what that sentence says. I thought that the degree of spread would be something along the lines I described above in STD example. 109.65.144.222 (talk) 06:00, 26 May 2014 (UTC)
 * Not sure it answers your question, but anyway: the normal distribution has two parameters, the mean and the spread; in contrast, Poisson distribution has only one parameter; its STD is necessarily the square root of its mean. Boris Tsirelson (talk) 07:02, 26 May 2014 (UTC)
 * Well, if that's the case, I'd say that the wording of the "it predicts the degree of spread around a known average rate of occurrence" isn't good. In a sense it always predicts the same degree of spread. It's not even useful for predicting this quality of spread. Why not just say that it predicts the chance of a Poisson random variable to receive a given value? Because to me that's what it does... 109.65.144.222 (talk) 16:37, 26 May 2014 (UTC)
 * True, taken literally it is, of course, "the chance of a Poisson random variable to receive a given value". And this is not specific to this distribution; the same may be said about every other distribution. But on the other hand, a distribution (Poisson or other) determines (somewhat indirectly) all numeric characteristics of location (expectation, median ...), spread (mean square deviation, interquartile range, ...), asymmetry and whatever. Looking at probabilities you easily get an idea which deviation is a rare event and which is not. Boris Tsirelson (talk) 19:04, 26 May 2014 (UTC)

I happen to notice this discussion, and yes, the referred sentence better could read something like: it describes the variation of the numbers around the mean. Nijdam (talk) 09:43, 28 May 2014 (UTC)
 * That would work for me too. The other thing about this particular distribution being able to describe spread is that it is too trivial - it just tells you back what you've already told it. I wasn't concerned with rarity of events. The way I understand what degree of spread measures it is how good is my prediction. I.e. it tells me that I've found a function whose values are at some good distance from actual observed values, and the degree is thus some relative units measured from some baseline (say, when my prediction is always correct) to some possible limit (my prediction is wrong half of the time exactly). In other words, try to pose yourself a question: "If Poisson distribution measures the degree of the spread, then what is the numerical value of the degree of the spread given my previous example of Acme Realty?" 79.182.18.4 (talk) 11:47, 30 May 2014 (UTC)
 * Sorry, I cannot understand your phrases "I wasn't concerned with rarity of events" and "when my prediction is always correct". Even dealing with the normal (rather than Poisson) distribution, the only always correct prediction is "somewhere between minus infinity and plus infinity". Then you writes "my prediction is wrong half of the time exactly"! But this is the opposite attitude. The "half" measures rarity (and therefore you are concerned). Boris Tsirelson (talk) 15:31, 30 May 2014 (UTC)
 * Sorry it took me a while to reply. Let's simplify it and concentrate on the following: what is the numerical value of the degree of spread given the Acme Realty example?. I can explain what I mean by other things you quoted, but it will drive the discussion away. I'd rather concentrate on answering this particular question. Just to expand on my motivation for resolving this question: my background in statistic is that I took a semester-long course in mathematical statistic, which is not much, but I imagine that I should be able to understand the description of some not very complex statistical structure, especially so, I can understand how it works and what it does. Yet the sentence is completely opaque to me. It's as if someone chained a handful of words in a syntactically valid way, but lacking any meaning. It is this meaning that I'm trying to discover, or to discover that the meaning was lost due to the bad wording. 79.176.121.21 (talk) 12:55, 7 June 2014 (UTC)
 * OK, now I see; you do not like the generic term "degree of spread"; instead you want to see something specific (be it mean square deviation, interquartile range or whatever). As for me, I interpret this "degree of spread" as a vague idea that hints to all these specific statistics, and to the vague intuitive idea of spread as well. However, the phrase in the article is not my formulation, and I am not really defending it. If you can write it better, be bold. Boris Tsirelson (talk) 14:54, 7 June 2014 (UTC)

on the CDF
Bikasuishin just updated the CDF to remove a leading 1 minus that was added by 128.244.208.67 about a week prior. So that after Bikasuishin's edit, it is.
 * $$\frac{\Gamma(\lfloor k+1\rfloor, \lambda)}{\lfloor k\rfloor !}\!\text{ for }k\ge 0$$

Since this appears to be a point of contention, it is worth noting that (using the definition of the Incomplete gamma function and using only integer values of k)
 * $$\frac{k! \ e^{-\lambda} \sum_i^{k} \frac{\lambda^i}{i!}}{k !}$$

which simplifies to
 * $$e^{-\lambda} \sum_i^{k} \frac{\lambda^i}{i!}\ $$

Isn't this a vastly simpler CDF function. In deed, one can easily see that it is increasing in k and even calculate it without clicking around to other pages. PDBailey (talk) 23:37, 8 February 2009 (UTC)
 * That's arguably simpler, but it's a lot less useful to the reader who wants to check a quick way to compute the CDF in a practical situation (which, by the way, is the reason I came to this page). It's much more efficient to use the representation in terms of the incomplete gamma function than the discrete sum. Convincing oneself that the corresponding formula is correct is a simple matter (provided one knows partial integration, at any rate). In case it's not immediately obvious, we can provide a derivation of the "gamma" version, but that's the one that belongs in the infobox. Bikasuishin (talk) 23:56, 8 February 2009 (UTC)


 * Both are CDFs and correct, but I am not really sure how one can integrate a function that is only non-zero at singularities without some familiarity of at least one of the non-Riemann integration concepts. The sum and floor function has the advantage of not requiring that in addition to being immediately derivable. I would propose that the CDF used for fast calculation needs a note in the text, not the other way around. PDBailey (talk) 04:23, 9 February 2009 (UTC)


 * What do you mean by "only non-zero at singularities"? You can't find much smoother than a function like tke-t, and the integral is certainly a convergent indefinite integral in the Riemann sense. But that's beside the point. To compute the incomplete Gamma function, you don't write a numerical integration routine (that's not the proper way to do it anyway). You do the same as you would for the error function: you fire up your favorite math software package (SAGE, Maple, whatever). Bikasuishin (talk) 09:56, 9 February 2009 (UTC)


 * What is the pdf of the Poisson at any point in $$k \in (3,4)$$? also, R, for example, does not have a built in incomplete gamma function. You could back one out (at the integers) using the CDF of the Poisson though. PDBailey (talk) 13:59, 9 February 2009 (UTC)

The leading "1-" is correct. A cumulative distribution rises steadily from 0 to 1. The Gamma function peaks at 0, so the present form is impossible!
 * This seems to be a confusion due to two extant definitions of the incomplete gamma function; see the upper and lower definitions at incomplete gamma function. If we are consistent and use a capital &Gamma; for the upper function, then the "1-" is indeed required. McKay (talk) 03:54, 5 September 2014 (UTC)

Introductory example
I think the introductory example is poorly explained/chosen.

The important thing about receiving mail that is not highlighted in the text, but that is the feature that makes mail per day _likely_ to be Poisson distributed, is the fact that each mail event occurs, to a good approximation for most people, independently of each other mail event. Saying that 'assuming the process or mix of processes that produces the event flow is essentially random' is, in my opinion, too vague and unclear. If we're taking about a probability distribution then of course we're talking about something that is 'essentially random'. What is likely meant here by 'essentially random' is independence of waiting times between events, but this could be stated explicitly. As it reads currently it sounds like the most important thing that makes mail/day a Poisson process is just that it's a random process described only by its average rate. But this is of course not the only distribution on the natural numbers that could be described only by its mean. Take waiting for a bus versus waiting for a taxi. Both taxis and buses may happen to pass you 4 times an hour, on average, but the distribution of 'taxis/hour' is much more likely to be Poisson than 'buses/hour' (depending on the city you live in...).

I would suggest:

For instance, an individual keeping track of the amount of mail they receive each day may notice that they receive an average number of 4 letters per day. It is reasonable to assume that receiving one piece of mail does not affect the arrival time of future pieces of mail, that pieces of mail from a wide range of sources arrive independently of one another, so the number of pieces of mail received per day follows a Poisson distribution. Other examples might include: number of phone calls received by a call center per hour, number of decay events per second from a radioactive source, number of taxis passing a particular street corner per hour.

Everybody knows this is nowhere (talk) 06:20, 13 September 2014 (UTC)

Law of rare events
In the explanation for law of rare events, the article claims that


 * "With this assumption one can derive the Poisson distribution from the Binomial one, given only the information of expected number of total events in the whole interval."

which is true; the distribution only depends on the expected number of total events in the whole interval and not the distribution of the probability of the events in the interval. However, the explanation then, goes ahead and assumes that the probability of an event happening in a subinterval $$I_i$$ is $$\lambda/n$$, which is only true if the events happen at constant rate in the interval, which defies the whole claim about we only need the expected value of total number and not the distribution of rate in the interval. --Sprlzrd (talk) 23:51, 29 April 2015 (UTC)

Graph suggests wrongly λ∈ℕ
The parameter λ in the distribution can take any positive real value, which is correctly stated in the table. However, in the graphs only examples with integer valued λ are given. This might lead to confusion, especially if someone visits the page with the thought "I think there were some restrictions on the distribution, something about only natural numbers", which might have been remembered because the outcomes of the distribution are natural numbers. Maybe whoever made the graphs could add an example of a non-integer λ? — Preceding unsigned comment added by 129.70.124.121 (talk) 14:49, 20 November 2015 (UTC)

Floor Functions in Summary Box
Since k is defined only over the set of natural numbers, the floor function of k or k+1 would never change the value. Should usage of and reference to floor functions in the function definitions in the summary box not be removed? — Preceding unsigned comment added by Antifrag (talk • contribs) 12:05, 14 December 2015 (UTC)


 * As written there, "The CDF is discontinuous at the integers of k and flat everywhere else because a variable that is Poisson distributed takes on only integer values." That is, the article follows the general definition of the CDF (for arbitrary distributions). For its restriction to natural numbers, of course, the floor function would not be needed. Boris Tsirelson (talk) 13:17, 14 December 2015 (UTC)

Homogeneisation
Sometimes the law is denoted by "Poi" and sometimes by "Pois", which is not homogeneous. Also, I am not sure this is the standard notation. 31.39.233.46 (talk) 10:51, 18 July 2016 (UTC)

Do we need the "Computer software for the Poisson distribution" section?
Do we need the "Computer software for the Poisson distribution" section? Realistically it is a very common distribution and effectively any generic statistical computing package will include this functionality. As it article stands there is little reason not to add MATLAB, SAS or even C (through some C external library akin to SciPy, like GSL) examples. Just something like: "Poisson distributed random variables can be readily generated in R, Excel, Mathematica, SAS, Python (SciPy), MATLAB, C/C++ (GSL), etc. ..." should be adequate. — Preceding unsigned comment added by Homo Ex Machina (talk • contribs) 00:23, 23 March 2017 (UTC)

Rate vs. average rate
The prerequisites section includes this item "The rate at which events occur is constant. The rate cannot be higher in some intervals and lower in other intervals." But isn't the whole point of this to calculate the probability of a higher or lower rate in each interval given the average rate across intervals? 66.62.244.4 (talk) 19:37, 3 October 2017 (UTC)


 * No. Probably, by rate in an interval you mean the number of events in this interval divided by the length of the interval. This is a random variable, while rate is nonrandom. This random variable relates to rate in the same way as frequency to probability. Frequency is random, probability is not. The expectation of the frequency is the probability. Similarly, the expectation of "your" random variable is the rate. Boris Tsirelson (talk) 21:02, 3 October 2017 (UTC)

"100-year flood" incorrect example
In the Examples section [], the first example is of a river that floods every 100-years. While the mathematics on this are fine, it perpetuates a misunderstanding. Apparently, scientists and engineers use the term "a 100-year flood" to mean not that an event happens once every 100-years, but that there is a 1-in-100 chance that it will happen in *any* year. (I was surprised, too, but here's a reference: https://www.nytimes.com/interactive/2017/08/28/climate/500-year-flood-hurricane-harvey-houston.html). Ben (talk) 08:46, 26 October 2017 (UTC)


 * Well, I think they do so in the context when the small difference between these two interpretations may be ignored. For "a 2-year flood" the difference would be notable. Boris Tsirelson (talk) 08:59, 26 October 2017 (UTC)

Incorrect statement in the intro
"expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known constant rate" Shouldn't this be something like a known mean rate?

Ma-Ma-Max Headroom (talk) 06:06, 2 November 2017 (UTC)


 * What do you mean by the "mean rate"? The rate $$\lambda$$ is not random (since the expected value of a random variable is not random), and not time dependent (thus, constant). Boris Tsirelson (talk) 09:25, 2 November 2017 (UTC)

I think he refers that while reading that statement one could understand that the events occur at a constant rate, in the sense that they're not a random process. It's as they where occurring one after another with a fixed period between them. Obviously one wouldn't need a probability distribution to describe that since it is not a random process. Anyways, it would be clearer if the statement said "if these events occur with a known constant mean rate". Because what is it constant is the mean value of the rate, not the rate itself. It's not clear in the statement that you're referring to an expected value. — Preceding unsigned comment added by 130.54.130.234 (talk) 06:33, 19 April 2018 (UTC)

Lomnitz citation
About the citation needed for (Lomnitz, 1994), this could be a valid substitute: (pdf) A review of earthquake occurrence models for seismic hazard analysis. Probably the original citation was intended for a chapter inside Fundamentals of earthquake prediction, but I don't have the book and I can't find any reference at that chapter anywhere.

I'm not that informed in this subject to make this decision alone. Stud94 (talk) 21:36, 22 May 2018 (UTC)

Rate (lambda) of the distribution
I read in the page "λ is the expected number of occurrences, which need not be an integer". This seems incorrect to me since Poisson deistribution is a discrete probability distribution. λ is the rate of the events and so it needs to be an integer. I've modified this but my changes have been reverted. Can someone explain me why this is not correct? Damiano Azzolini (talk) 20:08, 9 April 2019 (UTC)
 * Per various parts of the article:
 * λ is the average number of events per interval
 * an event can occur 0, 1, 2, … times in an interval
 * As an example, if there are 5 events that occur in 2 intervals, then λ=$5/2$. So, λ need not be an integer, but it must be a real number. Mind  matrix  20:31, 9 April 2019 (UTC)
 * And to clarify, the number of events in each interval is an integer. In my example, it could be 4 events in the first interval and 1 event in the second. Mind  matrix  20:33, 9 April 2019 (UTC)

Still missing: algorithm to draw from the distribution
Still missing here: reference to and brief explanation of algorithms for drawing random integers according to the Poisson distribution. -- Oisguad (talk) 19:22, 26 April 2019 (UTC)

Does the mean have to be a real number
Can the Poisson Distribution apply in a sitation where the mean has to be an Integer? Or is it a necessary condition that the mean is a continuous number? (I realise the samples are always integer, but I'm wondering about the mean)

I can't actually think of an example and that may be an answer in itself! BioImages2000 (talk) 09:55, 30 May 2020 (UTC)

What if lambda is not an integer, and is negative?
I've got a bunch of datasets which look like a Zipf distribution but with an exponentially-decreasing tail. See, for example, the bottom-most graph in Does Wikipedia traffic obey Zipf's law? which is a Poisson distribution with $$\lambda=-1/2$$. What's the "physics" of this? So, for radio-active decay, Posson is the continuum limit of many independent decay events. For page views, it is the, uhhh ... continuum limit, of, uuh ... things people are interested in? But these are not uncorrelated: people are more interested in "popular" things, be definition of the word "popular". So ??? 67.198.37.16 (talk) 01:54, 2 February 2021 (UTC)

Edit war about the PDF
There is something of an edit war 2021-11-11 regarding the correct PDF for the Poisson distribution. Someone is changing it to the wrong $$\frac{e^{\lambda} \cdot \lambda^{-k}}{k!}$$, even though it is obviously wrong. Don't know what to do about it. — Preceding unsigned comment added by 89.253.76.62 (talk) 19:47, 12 November 2021 (UTC)


 * This is a form of vandalism and it is serious since it's against the definition of a mathematical object. It has been going on for about a week now, I think we should consider semi-protection. 2021-11-17

Semi-protected edit request on 29 April 2022
Add the countable additivity property of independent Poisson distributed random variables. Makkar Aditya (talk) 17:34, 29 April 2022 (UTC)
 * Red question icon with gradient background.svg Not done: it's not clear what changes you want to be made. Please mention the specific changes in a "change X to Y" format and provide a reliable source if appropriate. ScottishFinnishRadish (talk) 17:37, 29 April 2022 (UTC)

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