Talk:Negative binomial distribution

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I can barely read any German, yet the article here made more sense than this one...74.59.244.25 (talk) 03:26, 18 February 2008 (UTC)

This article needs a proper introduction, that can help a layman understand what the term means, what it entails when used in text or conversation. Currently, this is not feasible, you'd have to scroll down a long ways and start reading the examples to even begin to understand; if you had no previous knowledge of mathematics or statistics at all. I'm putting this at the top, as I think it's more vital issue than any concerning the mathemathical/statistical content of the page. Starting with

In probability and statistics the negative binomial distribution is a discrete probability distribution.

Does not explain what Negative Binomial Distribution is - what separates it from other discrete probability distributions. I personally think this should be attempted as highest priority, obviously I'm not able to do it (or I wouldn't be writing here, eh). Assuming anyone (not to mentione everyone) is able to understand mathematical formulas that incorporate greek letters is IMHO pedagogically unsound --Asherett 12:17, 13 September 2007 (UTC)

Well, obviously the statement that "In probability and statistics the negative binomial distribution is a discrete probability distribution" does not way WHICH discrete probability distribution it is---that comes later in the article. As for making it clear to someone who knows NOTHING AT ALL about mathematics or statistics: that may not be so easy. Perhaps making it clear to a broader audience can be done, with some effort, though. Michael Hardy 19:15, 13 September 2007 (UTC)

I have reverted the most recent edit to negative binomial distribution for the following reason.

  • Sometimes one defines the negative binomial distribution to be the distribution of the number of failures before the rth success. In that case, the statement that the expected value is r(1 &minus p)/p is correct.
  • But sometimes, and in particular in the present article, one defines it to be the distribution of the number of trials needed to get r successes. In that case, the statement is wrong.

If you're going to edit one part of the article to be consistent with the former definition, you need to be consistent and change the definition. Michael Hardy 17:40, 7 Jul 2004 (UTC)

Contents

[edit] Equivalence?

If Xr is the number of trials needed to get r successes, and Ys is the number of successes in s trials, then

\operatorname{Pr}(X_r \leq s)=\operatorname{Pr}(Y_s \geq r).

The article went from there to say the following:

Every question about probabilities of negative binomial variables can be translated into an equivalent one about binomial variables.

I removed it. I tentatively propose this as a counterexample: Suppose Wr is the number of failures before the r successes have been achieved. Then Wr has a negative binomial distribution according to the second convention in this article, and it is clear that this distribution is just the negative binomial distribution according to the first convention, translated r units to the left. This probability distribution is infinitely divisible, a fact now explained in the article. That means that for any positive integer m, no matter how big, there is some probability distribution F such that if U1, ..., Um are random variables distributed according to F, then U1 + ... + Um has the same distribution that Wr has.

So how can the question of whether the negative binomial distribution is infinitely divisible be "translated into an equivalent one about binomial variables"? Michael Hardy 01:43, 27 Aug 2004 (UTC)

Removing the bit about "every question" seems OK to me; the important point is the relation between binomial and negative binomial probabilities. But Mike, it wasn't put in there for the purpose of annoying you. You might consider using the edit summary to say something about the edit rather than your state of mind -- how about rm questionable claim about "every question" instead of I am removing a statement that has long irritated me. Wile E. Heresiarch 15:23, 6 Nov 2004 (UTC)

[edit] Major reorganization

Trying to be bold, I've just committed several major changes. I found the previous version somewhat confusing, since it talked about three slightly different but closely related "conventions" for the negative binomial, and it never became fully clear to me which convention was in use at which point in the subsequent discussion. I've replaced the definition with what I consider to be the most natural version (the previous convention #3). The reasons that definition is "natural" is that it arises naturally as the Gamma-Poisson mixture, converges-in-distribution to the Poisson, etc. The shifted negative binomial (previous convention #1) can still be derived (see the worked example of the candy pusher). Now we have a single, consistent (hopefully!) definition of the negative binomial instead of three similar-yet-different conventions. I'm painfully aware that all of the previous three conventions are in use and sometimes referred to as the negative binomial; but then again, that doesn't even begin to exhaust the variations on this distribution that can be found in the wild, so why not pick one reasonble definition and stick to that here? --MarkSweep 12:04, 5 Nov 2004 (UTC)

Well, if we were writing a textbook, we would certainly want to pick one defn and stick to it. However, we're here to document stuff as it is used by others. If there are multiple defns in common use, I don't see that we have the option to pick and choose. Sometimes multiple defns can be collapsed by saying "#2 is a special case of #1 with A always a blurfle" and then describing only #1. I don't know if that's feasible here. Regards & happy editing, Wile E. Heresiarch 15:09, 6 Nov 2004 (UTC)
Yes, that was basically the case here. The previous "convention #2" was the Pascal distribution, which is a special case of the general negative binomial (previous "convention #3"). This didn't become fully clear in the previous revision, where the discussion of the Pascal distribution seemed more like an afterthought. The previous "convention #1" appeared to be simply a Pascal distribution shifted by a fixed amount. There is still a discussion of that in the worked example, but that could arguably be moved to the front and made more explicit. --MarkSweep 23:25, 6 Nov 2004 (UTC)
Hi, just found this page and I don't like that the starting point is the more general formula that has r being a strictly positive real. I think that 99% of the time somebody is interested in this distribution, r is going to be an integer. Which isn't to say that we should purge this more complete definition, just that there is a lot to be said for following the way the present article on the Binomial distribution is written (since this is closely related) and because that one is a heck of a lot clearer. I would suggest using one variable where r is an integer and a seperate variable where it is a real (to keep them straight). Along the same lines, I also think that starting talking about Bernoulli trials so far down the page is not a good idea--I'd like to see it up top. Is this what you two are talking about? Oh, wait, those dates are 2004! oh well, I'll still wait to see if anyone cares b/c this is a big edit. --Pdbailey 07:13, 9 November 2005 (UTC)
I support the previous comment. I am a graduating maths/computer science student, but the first definition was absolutely non-intuitive for me and only the "Occurrence" section made it clear. I doubt whether the generalization is more important than the fact that this distribution is derived from the Pascal distribution. —The preceding unsigned comment was added by 85.206.197.19 (talk) 20:10, 4 May 2007 (UTC).

[edit] Plots?

Is it possible to get some plots of what this looks like? I got sent here from the mosquito page, and anyone reading that probably doesn't want to wade through many lines of math, just see a picture of what it means. --zandperl 04:10, 30 August 2005 (UTC)

  • One year later, exactly the same issue. Remarkably, the mosquito page still links here, but there's no plot. Anyone?

Sketch-The-Fox 23:21, 19 August 2006 (UTC)

[edit] the mean is wrong

should be (1-p)r/p, surely


UM According to 'A First Course in Probability' by Sheldon Ross, the mean is r/p

Wrong. Look, how many times do so many of us have to keep repeating this? Sheldon Ross's book CORRECTLY gives the mean of what Sheldon Ross's book calls the negative binomial distribution. But there are (as this article explains) at least two conventions concerning WHICH distribution should be called that. Sheesh. Michael Hardy 21:42, 29 November 2006 (UTC)

[edit] the mgf is wrong

The numerator should be pe^t instead of p. The following link can support this http://www.math.tntech.edu/ISR/Introduction_to_Probability/Discrete_Distributions/thispage/newnode10.html

The bottom of that page gives the mgf of negative binomial distribution. I verified it. —Preceding unsigned comment added by 136.142.163.158 (talk • contribs)

WRONG!!! This article has it right, and so does the web page you cited. They're talking about TWO DIFFERENT DISTRIBUTIONS. You did not read carefully. The negative binomial distribution dealt with in this article is supported on the set
{ 0, 1, 2, 3, ... }
whereas the one on the web page you cite is supported on the set
{ r, r + 1, r + 2, .... }
Both articles are clear about this. You need to read more carefully. Michael Hardy 19:55, 13 September 2006 (UTC)

[edit] Use of gamma function for a discrete distribution

Is it the convention among probability literature to represent the negative binomial with the gamma function? In Sheldon Ross's introductory text, the distribution is introduced without it (although that is an alternative representation of the distribution). I am not objecting but as a beginner am curious why this is how it is represented. --reddaly

I think either adding this way of writing it: \binom{n-1}{r-1}p^{r}(1-p)^{n-r}, or specifying that Γ(x + 1) = x! would be beneficial. some people start running when they see the gamma function

Good idea. It would be easier on the eyes for those who haven't yet discovered how to love the Γ function. Aastrup 22:24, 18 July 2007 (UTC)

[edit] Expected Value derivation

The classic derivation of the mean of the NBD should be on this page, as it is on the binomial distribution page. --Vince |Talk| 04:44, 12 May 2007 (UTC)

I agree. Aastrup 22:24, 18 July 2007 (UTC)

[edit] MLE

This article lacks Maximum Likelihood, and especially [Anscombe's Conjecture]] (which has been proven). Aastrup 22:24, 18 July 2007 (UTC)

[edit] overdispersed Poisson

I recently added a note about how the Poisson distribution with a dispersion parameter is more general than the negative binomial distribution and would make more sense when one is simply looking for a Poisson distribution with a dispersion parameter. I think it's important to realize that the Poisson distribution with a dispersion parameter described by M&N is more general in that the variance has positive support instead of the more limited greater support than the mean. There certainly are situation where the negative binomial distribution makes sense, but if one is just looking for a Poisson with a dispersion parameter, why beat around the bush with this other distribution and not just go for the real thing? Pdbailey (talk) 17:38, 26 January 2008 (UTC)

There is no such thing as "overdispersed Poisson", because if it is overdispersed, then it is not Poisson. If "the Poisson distribution with a dispersion parameter described by M&N" is important, then go ahead and describe it in some other article, perhaps in a new article. This article is about the negative binomial distribution only. The (positive) binomial distributions have variance < mean, and the Poisson distribution has variance = mean, and the negative binomial distribution has variance > mean. Bo Jacoby (talk) 22:36, 26 January 2008 (UTC).