Talk:Probability distribution

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Mathematics grading: B Class High Importance  Field: Probability and statistics
Needs less technical introduction; more prose to accompany many of the items on lists; and information on history. Tompw 18:29, 7 October 2006 (UTC)

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[edit] CDF vs. PDF

For Continous Random Variable you ARE giving the Cumulative Distribution Function, not the Probability Density Function...eh?

The cdf is defined for all random variables, discrete or continuous, so it is a better starting point than either the probability function or the density function. In one case you use differences to get the probability function and in the other you use the derivative. Most students are introduced to the derivative before the integral, so this approach is a bit more accessible -- DickBeldin

The probability that a continuous random variable X takes a value less than or equal to x is denoted Pr(X<=x). The probability density function of X, where X is a continuous random variable, is the function f such that

  • Pr(a<=X<=b)= INTEGRAL ( as x ranges from a thru b) f(x) dx.

Correct, but F[b]-F[a] gives the probability of an interval directly without all the complications. We hide the complications in the cdf. It is inconvenient that we can't feature the explicit form of the cdf for many of the distributions we like to use, but it is important to build the concepts with proper spacing of the difficulties. One hurdle, then a straight stretch, then a curve, then another straight ... --DickBeldin----

You may present this material as you feel best. I don't disagree with your argument. But, mislabelling definitions is never okay. You have defined the probability density function for continuous random variables with the cumulative distribution function for the same. RoseParks

Surely you mean absolutely continuous. And defining the pdf from the cdf the right way to do things. If you want to go to first principles, you need to specify a Borel measure on the real numbers, and the best way to do that is using a Lebesgue-Stieltjes measure. In probability theory, you call measures distributions and the Lebesgue-Stieltjes measure is called the cdf. -- Miguel

[edit] Restriction to real-valued variables

The definitions given on this page seem much too limited. A probability distribution can be defined for random variables whose domain is not even ordered (take the multinomial, for instance). In these cases, the cumulative distribution function makes no sense. To claim, as this page does, that the distribution must have the reals as the domain is nonsense.

Agreed, but it seems to be customary to use this restricted interpretation of the domain. It is possible to define the cdf for vector-valued random variables (incuding your example) but this is very clumsy. Vector-valued functions are usually treated as collections of correlated real variables. -- Miguel

[edit] The Boltzmann distribution

The so-called Boltzmann distribution is a strange beast to include in the list of discrete distributions, as are all the "special cases" listed under it. The reason is that the Boltzmann distribution is just a rule that, given a collection of states (not necessarily a set of real numbers) and their energies (not necessarily all distinct) gives a probability measure on the collection of states. It can be applied to discrete and continuous collections of states, and especially in the discrete case there is no reason why the states should be labelled by real numbers. Some of the special cases, for instance the Maxwell-Boltzmann distribution, are not even discrete! — Miguel 21:32, 2004 Apr 24 (UTC)

  • All true, but it is still an important distribution. The fact that it has a strong relationship to physics does not single it out. --Pdbailey 01:35, 1 Sep 2004 (UTC)

[edit] Rare events

I think calling the Poisson and associated "counting" distributions as regarding 'rare random events' is not quite right. For example, counting decayse of Potassium-40 with a gamma spectrometer, one could count a hundred per second... I would edit it, but I can't come up with a better way of saying it. Can you?--Pdbailey 01:32, 1 Sep 2004 (UTC)

Nonetheless, they are rare in the sense intended. The reason they are Poisson-distributed is that there is only one decay out of each zillion or so opportunities. Michael Hardy 14:18, 2 Sep 2004 (UTC)
... and besides, if you were looking at some very long time -- say several seconds -- you'd probably want to model it as a normal distribution rather than as a Poisson distribution. Michael Hardy 14:19, 2 Sep 2004 (UTC)
Well, I understand what you are saying, but the point is to make it easier to understand. If I get 2 counts per second, then after 30 seconds, the normal distribution isn't going to do it. But the events are not rare. Me washing my car is rare, something that happens at 2 Hz is not rare. Pdbailey 05:17, 4 Sep 2004 (UTC)
The original meaning of rare in this context was that the probability of two events occurring simultaneously is zero. You may disagree with that use of the word, but we're stuck with it for historical reasons. I have heard that one of its first uses in the 19th century was to model the occurrence of army officers being kicked by horses.
Also, it becomes clear in what sense the Poisson distribution is rare if you look at its derivation as a limit of the Binomial.
Finally, whether two decay events per second are rare depends on the timescales involved. For you a second seems like a very short time, but when talking about subatomic physics a second is an eternity. Similarly, you think washing a car is a rare event bacause you measure the frequency per day, or per week. If you do it per year it ceases to be "rare" by your definition.
There are three time scales involved here: the time resolution of the experiment; the average time interval between events; and the unit of time used to express frequencies. The unit is irrelevant. If the resolution is much smaller than the interval, you use Poisson. If it is much larger, you use normal (as an approximation: Poisson is still exact). — Miguel 16:25, 4 Sep 2004 (UTC)
Your arguments (refered to by paragraph) do not hold water. (1) The defintin of rare is well given by the wiktionary as "very uncommon; scarce", and how somebody misued it centurys ago while describing this perticular distribution does not change the fact that this is misleading. (2) it becomes clear not the it is rare but that each iota of the Poisson distributed events is unlikely, but not rare. Phone calls ariving at a help center will often be frequent (think many per second) and are Poisson distributed. The chance that any given person called is very low. (3) we can dismiss this out of hand with the previous example.
The Poisson distribution is the limit of the Binomial distribution when the probability of success goes to zero (hence rare events) but the average number of successes per unit time is kept constant when taking the limit. Hence, rare events. If you don't get it, you don't get it.
There is such a thing as historical accidents, conventions and tradition in the way science, technology and all of human knowledge is organized. You have to live with that, and wikipedia is not the place to revolutionize notation or terminology. If you don't get it, you don't get it.
Miguel 03:02, 7 Sep 2004 (UTC)
Miguel, please explain to my how a call center that receives 20 calls per second is observing rare events? Read the definition, "scarce." 20 calls per second is hardly a drought. What I have changed it to, "which describes a very large number of individually unlikely events" is more accurate (captures the derivation from binomial distribution). If you have another wording that is more accurate, please, propose it. Pdbailey 04:43, 7 Sep 2004 (UTC)
I told you the unit in which you measure time is irrelevant. You are talking about .05 calls per millisecond.
Now seriously, your change is ínaccurate because the Poisson distribution can describe a very small number of individually unlikely events, too. The name "distribution of rare events" is something we're stuck with for historical reasons, and it is a synonim for "Poisson distribution". try this:
The Poisson family of distribution describes rare independent events and is parameterized by the average number of events occuring. Note that the average number of events can be large or small depending on the situatin, and that it is the individual events that are "rare".
Here's the problem: the Poisson distribution is as ubuquitous as the normal distribution and has meny applications. The intuitive explanation why the Poisson distribution applies in one particular situation may be "misleading" in another situation. The list of probability distributions is not the place to discuss those nuances, that's what the article Poisson distribution is for. — Miguel
Miguel, please answer this set of questions. Rare is a relative word, if you want to be clear, almost any other word would be better. Please expalain why you want to use it. You keep saying that we are stuck with it for historical reasons. What is your argument? why do we have to use it based on 'historical reasons.' What are the historical reasons?
We are stuck with it for reasons of tradition. That is the name it was given in old texts, and such things propagate as people copy each other. — Miguel 17:17, 2004 Sep 12 (UTC)
I do not like your definition because it is overly wordy, "note...can be...depending...and that it is..."
My definition is very tight and accurate, let me argue for it.
which describes a very large number of individually unlikely events that happen in a certain time interval.
This is inaccurate: it can describe a very small number of events, too. Can't you see that the number of events (per unit time) can be any positive number, large or small? Can't you see that simply changing the unit of time can make this average as large or small as you please? — Miguel 17:17, 2004 Sep 12 (UTC)
Poisson distributed events must be a large number (look at the proof on the page for the Poisson distribution) each of which must be unlikely (look at the proof on the page). The time interval bit diferentiates it from the Erlang distribution. Again, this definition is short, accurate, and even hints at the proof.
Finally, this page (discussion) is for the discussion of entries on this page. So long as the sentence is on this page, discussion about it belongs here.--Pdbailey 00:10, 11 Sep 2004 (UTC)
Do as you please, I don't care any more. — Miguel 17:17, 2004 Sep 12 (UTC)

[edit] other distributions

Are the Rayleigh distribution and Rician distribution important enough to be included in the List? -FZ 15:13, 6 Jan 2005 (UTC)

What about the Nakagami-m distribution? -Mangler 12:05, 26 July 2005 (UTC)

[edit] Zipf and Zipf-Mandelbrot

Zipf's law is for a finite N = number of elements or outcomes, for example, the number of words in the English language. When N becomes infinite, Zipf's law becomes the zeta distribution. Zipf-Mandelbrot law is a generalization of Zipf. When N becomes infinity, Zipf-Mandelbrot becomes something, I don't know what at this point, but whatever it is, it involves the Lerch transcendent function just as the zeta distribution involves the Riemann zeta function. PAR 08:04, 12 Apr 2005 (UTC)

You're right. Sorry about moving it back – I had looked at the support field in the infobox for Zipf's law and decided to recategorize the article here. The right thing to do for me would have been to fix the support field, which suggested that k ranges over the full set of natural numbers. I'm gonna work on that now. --MarkSweep 13:58, 12 Apr 2005 (UTC)

Ok, I'll go ahead with some plots for Zipf and zeta. PAR 15:06, 12 Apr 2005 (UTC)


[edit] The Economist

This page was featured in The Economist at Psychology - Bayes Rules.

From: Schaefer, Tom PAX Tecolote Sent: Monday, May 08, 2006 9:24 AM To: 'robert.dragoset@nist.gov' Cc: McDowell, Jeff HSV Tecolote Subject: Uncertainty in Physical Values

Hi PhD Dragoset,

   I hope your presentation on units to OASIS went well, and the issue is being given the priority and urgency it deserves.

   The input values required for the execution of our cost estimating models are often uncertain, and rather than a discreet value, can be more accurately described as a range or distribution of values.  I would like you to consider helping to develop an XML standard for representing the uncertainty of a numeric value in a standard way that “Monte Carlo” and other analysis tools could interpret universally.  The intent would be to develop an XML element that could be used almost anywhere a quantitative attribute is currently used (the current discreet or point value being a subset of the element).  The parameters for common distribution types, such as uniform, triangular, Gaussian, beta, Poison, Weibull, etc. ( http://en.wikipedia.org/wiki/Probability_distribution ) would be supported, as well as a way to represent a data set of values to sample from.

   Do you have any interest in this topic?  It could have profound importance to the meaningfulness of data exchanges and quality of analysis.

Tom Schaefer Senior Technical Expert Tecolote Research, Inc.

[edit] Diagnostic Tool

I think there should be a section on randomness as a diagnostic tool in certain mathematical applications, such as regression etc. Just a thought... --Gogosean 20:58, 15 November 2006 (UTC)


[edit] Plots

The plots in general are quite illustrative and pretty. However, for the discrete distributions (Poisson, etc.) wouldn't it make more sense to have bar-like plots rather than connected points? The segements between the points have no meaning as far as the distribution is concerned; only the frequency value does. I understand it's tricky to superpose bar graphs, but there are ways around that, like outlining the bars or making separate plots. Also, the order of the eplots seems arbitrary. Why is the relatively obscure Skellam distribution near the top and the Gaussian at the very bottom? Shouldn't the beta and the gamma be closer? Shouldn't the t and F distributions be included? Or the binomial and negative-binomial? -- Eliezg 01:49, 9 December 2006 (UTC)

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