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If you go to previous versions and look at the first one, 02/15/2001, which is yours?, you will see :
1). q (1-p), maybe a typo?
2). And the formula for the numbers of ways of picking X items out of N items was:
N!/X!/(N-X)!. This is plain wrong. Yes, after requesting a change for a week, I changed it.
3).There were also wording problems. RoseParks.
I see now the problem. (1-p) was intended as a parenthetical definition. I guess N1/X!/(N-X)! worked in my programming codes so I couldn't see the ambiguity. How would you calculate N!/X!/(N-X)!? From right to left? On the other hand, Today is 02/20/2001, so I think your "requesting a change for a week" is a bit off. Today is only the 20th by my calendar. In any case, the criticism has led to something better. Dick Beldin---- In answer to your question on how you evaluate, N!/X!/(X-N)!, this is ambiguous. In any easy example.
2/4/12 is ambiguous since
Multiplication is associative over the reals. If you look at division as the inverse operation of multplication, i.e. 2/4/12=2*4^1*12^1=1/24 you are okay. If you look at division in the ordinary sense, you must specify the order of operations.RoseParks
I agree that an expression with successive divisions appears ambiguous. Most mathematicians I know do indeed consider division as the inverse of multiplication and many programming languages explicitly specify that multiplication and divisions are performed left to right. You are correct, it is not a universal convention. In addition, the vertical placement of numerator and denominator is clearer. Dick Beldin
I was looking for information about confidence intervals on a binomial distribution, but was surprised not to find it here. I know this case isn't quite as simple as for normal distributions, but it would be nice to have here, if somebody would like to contribute the information.
You mean CI of p, the success probability, as estimated from the data. If 70 successes in 100 trials, then p_est = 0.7, and your question is what is standard deviation of p_est. It is sqrt(p_est(1-p_est)/n_trials). The 95% confidence interval is +/- 2 standard deviations. My question is what happens if the CI range is outside the allowed 0 to 1 range for a probability. This can happen if p_est is ~1 or ~0. The CI has to be assymetric. Any ideas?
I was looking for a pointer to quickly simulate a Binomial trial. That is, given a p and an n, I want to randomly select a result with a Binomial distribution. I know I can approximate this with a normal distribution, but I would prefer an exact result if it can be calculated quickly for n < 10,000. I'm sure others have come here looking as well. Thanks.
Is it me, or should the "A typical example is the following: assume 5% of the population is HIV-positive." part in the second paragraph be changed to something a little less... you know... The HIV part is just not encyclopedia-ish...
Okay, maybe this is standard jargon somewhere, but I've never come across it until today. I guess "mass" makes sense by the physical analogy to density. Honestly, I think it's stupid language. Should we also speak of cumulative mass distribution functions? Be consistent! I'm not going to change it, but a mathematician should. At the very least link it to the pmf page.
The article gives the following example: "A typical example is the following: assume 5% of the population is green-eyed. You pick 500 people randomly. How likely is it that you get 30 or more green-eyed people?".
This is a CDF example. Unfortunately, the expression given for CDF is not very clear to me. How about giving a worked example with the green-eyed people given in the article as a good example, please? --New Thought 15:12, 8 May 2006 (UTC)
BetaRegularized[0.05, 30, 471]
. Direct summation is likely going to be less numerically stable than a carefully designed subroutine for evaluating the incomplete Beta function. --MarkSweep (call me collect) 06:11, 9 May 2006 (UTC)I really dislike the "nmemonic" section. If anyone else agrees, please delete it. McKay 14:55, 11 June 2006 (UTC)
How about putting it here, then with attention on it someone may come up with better. Tabby 03:44, 31 October 2007 (UTC)
Here is the diff: [1]. But I agree with deleting it, it is unencyclopedic. Sander123 13:58, 31 October 2007 (UTC)
The article currently states: The formula for Bézier curves was inspired by the binomial distribution.
Would someone care to source that statement? It seems rather dubious to me, but if it's true it's worthy of a proper explanation and not the vague description of being "inspired by". Certainly the Bernstein polynomials, which constitute the basis functions for Béziers, contain a Binomial coefficient. But binomial coefficients exist all over the place. It doesn't necessarily imply that they have much at all to do with the Binomial distribution.
From reading about Bézier curves I've always had the impression that the decision to use Bersteins as their parametrization wasn't 'inspired' by anything, but merely chosen from a group of candidates on the merit of their desireable properties. (Being such properties as the fact that curve is guaranteed to be contained within the convex hull of the control points, that reversing the control points does not change the curve, that the tangents at the endpoints consist of the line between the endpoint and the neighboring control point, etc). --130.237.179.166 14:48, 3 September 2006 (UTC)
I feel like there could be a better example than picking 500 people out of a population "with replacement" and seeing how many were green-eyed. Perhaps a more sensical and applicable example could be: out of 50 web servers, each of which has a 1% chance of failing by the end of the day, how many failed servers do you have at the end of the day?
—The preceding unsigned comment was added by 18.216.0.100 (talk • contribs) .
The example with a die is OK in principle. Most people, I reckon, will have seen and used dice. But why change the well known configuration ( 1 thru 6 dots) with "5 blank and 1 black side"? This now makes a familiar object unfamiliar and thus more difficult to mentally latch onto. Furthermore, a lesser issue, 'blank' and 'black' are two very similar words possibly leading to misreading. Why not just simply use: "Roll a die ten times and count the number of sixes.", thus appealing to a general feeling of wishing to see the highest value side turn up? Gerald Tros 01:41, 21 May 2007 (UTC)
--Why not start with a coin example? Isn't this the most straight-forward? The one that everyone did in 4th grade??--128.135.96.223 (talk) 20:34, 9 March 2008 (UTC)
-- Why does the article use a biased coin in the introductory example? Why not a normal coin that people encounter in a normal life? Bosons (talk) 16:49, 19 January 2017 (UTC)
I deleted a new section on "Kitchen's theorem". It began by saying "...we can see by Kitchen's Theorem that..." without having first said what "Kitchen's theorem" is. That is not appropriate. Then, as far as I can tell, the theorem turned out to be a proposition found in many textbooks without the name "Kitchen's theorem". The notation in which it is written includes the use of the same letter for two different random variables in the same equality. Near the bottom it has some notation that is less than correct and that includes some very clumsy language. Then there is a signature---appropriate for a talk page but not for an article. In includes "Dr. William Kitchen PhD (Psychology)", apparently identifying that person as the one who added this material. It looks like an attempt to name after himself a proposition found in innumerable textbooks since before the births of most (or all?) people now living. Michael Hardy 20:11, 23 March 2007 (UTC)
Well Michael, it's nice to see a fellow 'Mathematician' scrutinising my work, labellng it a 'proposition'. Given the fact that my Theorom has went under rigorous investigation within a university, I fail to see how you can ever have seen it in "innumerable" textbooks. Perhaps you could name a few of them for my reference. And lets not get into a Mathematical jargon slanging match; whoever you are, I would be confident in my own Mathemaical standing to stand before anyone and prove my Theorem/ lemma. And, if it is indeed in many textbooks, I'd urge you to publish a proof of my statement. I have it on good authority, from highly esteemed Mathematicians, that the Theorem I put online is indeed a new and may I add correct proposition. It wasn't a Theorem as such, hence why I referred to it as the Binomial Lemma. I trust you know what a Lemma is! In future, before you make such claims, ensure that the nature of your statements is true. Do that rather than correcting me. And in response to this, if you do indeed give one, I'd appreciate being referred to as Dr. William Kitchen.— Preceding unsigned comment added by 84.66.3.105 (talk • contribs) 18:00, 29 March 2007
What you quoted from this textbook isn't even the same as my Theorem. And do not quote Wikipedia policy to me - take me to court, sue me, do whatever you wish. I have this Theorem in a journal, and have had it copyrighted to my name, so that scavengers on internet sites cannot attribute a novel idea to a text book they happen to have read. I had it checked, along with a proof by a university Professor who specialises in the concpets of probability and statisitics. It then underwent a stage of 'gaining plausibility', and under futher rigorous proof. There was a work through proof, and a proof by induction which clearly shows that the NEW theorem works, for all the possible values it outlines. I think you'll find the quote you have from your book involves a different concept to what I outlined before. I'll tell you what : take a look at it, and as Fermat said before he published his last Theorem "prove me wrong": I've got a mortgage on it saying you can't!! All the best, Dr. William Kitchen — Preceding unsigned comment added by 84.66.3.105 (talk • contribs) 22:59, 29 March 2007
Well, I appreciate that. Like all Mathematicians, I like recognition for my work. I had to have it rigorously checked and compared with similar Theorems and Lemmas, to ensure I wasn't putting my name to a piece of work that someone else had previously discovered. Notation is a blunder, I hold my hands up on that front, and I understand the elementary nature of my error. I can provide you with my proof for the Theorem as soon as I finish my textbook which is in finalisation at the moment. All my work is momentarily on hold becasue of that. I welcome any scrutiny of my work - I feel that Mathematics is best done when under pressure from other esteemed Mathematicians. The workings of Wikipedia, however, are something I am not aware of, and I appreciate any guidelines you offer me to follow. Again, however, as I have already said, I know I can stand before any Mathematician and prove my Theorem. Regards Dr. William Kitchen
Dr. Kitchen, could you tell us the title of the paper and the name of the journal and which issue it's in?
That would really be a whole lot more to the point than telling us how confident you are that everything about it is sound. Michael Hardy 20:31, 30 March 2007 (UTC)
Not sure about the statement
The exact calculations are only onerous if one doesn't have a computer. Considering that virtually all statistics is done over computers these days the above seems unimportant. 128.195.106.28 23:55, 31 March 2007 (UTC)
Hmm, just a reader here, but I can't make a modern computer delay visually for any reasonable n (up to 9999999999) when using the exact solution. I advise my students to always use the exact test and that the normal approximation is a relic of a bygone era. However it is interesting and perhaps worth noting why the binomial becomes normal-ish. Also, I thought the ability to use the normal approximation was based on np not n - with a low enough p, even a huge n will be skewed.4.79.81.6 04:45, 1 November 2007 (UTC)
I experienced, that the normal approximation is indeed a time-saver if e.g. computing many different binomial distributions. In my case -- using octave -- computation speeded up a lot, especially since I was using quite large n's and always had to sum up about n/2 distributions (for only one point in the plot). So thank you for mentioning it in the article! --129.13.186.1 (talk) 10:13, 18 September 2009 (UTC)
Your end result for the binomial approximation is incorrect. It should be N(np , (np(1-p))^1/2). You currently have N( np , (np(1-p))). —Preceding unsigned comment added by 24.29.95.138 (talk) 21:59, 24 January 2010 (UTC)
This section is my first contribution. I sincerely hope it's sensible to have done so and that it is a (potential) boon to readers. I'm honing it, adding links, references, improving text etc. Please give me a couple of days, I'll post it in one single edit. I'd appreciate any advice you have for me regarding content choice, style etc. Thank you. Thanks already to Michael Hardy. Gerald Tros 01:34, 25 April 2007 (UTC). OK, a couple of weeks. It's almost ready :-) Gerald Tros 01:31, 11 May 2007 (UTC) Done. Gerald Tros 01:28, 16 May 2007 (UTC)
Might it not be a lot easier to demonstrate this proof using generating functions? I can easily do it this way, unless anyone can spot a good reason not to (it requires a lot less algebra...but does requires some GF results) Wrayal 20:45, 31 May 2007 (UTC)
The cdf of a discrete distribution must be piecewise constant. ПБХ 15:13, 21 September 2007 (UTC)
I hope somebody could help me in finding derivations or how to derive the skewness and kurtosis, even link to other sites will be much appreciated. —Preceding unsigned comment added by Student29 (talk • contribs) 19:29, 16 January 2008 (UTC)
After giving the expectation as np, the article states "This fact is easily proven as follows. Suppose first that we have exactly one Bernoulli trial. We have two possible outcomes, 1 and 0, with the first having probability p and the second having probability 1 − p; the mean for this trial is given by μ = p." This is not a proof. These sentences should really just be removed. —Preceding unsigned comment added by 68.50.194.132 (talk) 19:59, 16 February 2008 (UTC)
In the section entitled Mean, variance and mode, it isn't clear to me how the expression given follows from "Using the definition of variance, we have..." Should I try to find this in the entry for variance, figure it out from the problem statement, or use the definition of variance given just above? In any case I don't see how it follows.Telliott (talk) 11:43, 14 March 2008 (UTC)
The figures have unlabeled axes, making them pretty much useless. Can someone either introduce new figures, or edit the existing ones to have axis labels? 209.94.128.119 (talk) 01:30, 13 November 2008 (UTC)
The article for Hypergeometric distribution describes how it is used for sampling without replacement and states that Binomial Distribution is used for sampling with replacement. How is Binomial Distribution method used for sampling? Virgil H. Soule (talk) 18:11, 2 July 2009 (UTC)
Removed:
As another example, assume 5% of a very large population to be green-eyed. You pick 100 people randomly. The number of green-eyed people you pick is a random variable X which approximately follows a binomial distribution with n = 100 and p = 0.05 (strictly a hypergeometric distribution).
If it isn't strictly a binomial distribution, then it is a bad example.
In lieu of misusing a hypergeometric distribution as an example of a binomial distribution, perhaps add a section detailing the relationship and how they are similar and yet different? Madkaugh (talk) 00:42, 13 October 2009 (UTC)
The expression for the mode is incorrect. Imagine a Binomial distribution with p = 1.0 and n = 2, the expression for the model will return 3 while the true value is 2. —Preceding unsigned comment added by 86.165.211.190 (talk) 21:52, 1 November 2009 (UTC)
Michael Hardy (talk) 06:17, 25 November 2009 (UTC)
It doesn't make any sense for the parameter n to be 0. Most texts limit n to be a natural number. —Preceding unsigned comment added by 128.187.81.187 (talk) 21:03, 23 November 2009 (UTC)
It's common in Wikipedia math articles to discuss algorithms for computing quantities of interest. On this page it would be very helpful to have a discussion of computing the cumulative distribution function (CDF). The article does mention various methods that can be used in various circumstances, but this is an incomplete solution at best. For example, the article doesn't provide any guidance about choosing between (a) a combination of direct summation and the normal approximation and the poisson approximation, (b) a method based on the incomplete beta distribution, and (c) something else. A discussion of computing the CDF would be useful to a lot of people. ATBS 22:28, 30 November 2009 (UTC)ATBS —Preceding unsigned comment added by ATBS (talk • contribs)
The second rule of thumb for normal approximations looks suspicious. It can be written in the following form: use normal approximation whenever
In particular that rule claims normal approximation should not be used for any n when p = ½. So the sign should probably be reversed, and the factor n omitted? … stpasha » 22:41, 30 November 2009 (UTC)
There is an error in an example: σ = (p(1 − p)/n)1/2. Should be σ = (np(1 − p))1/2. Please someone fix it or explain what I do not understand there. —Preceding unsigned comment added by 213.197.179.210 (talk) 10:34, 5 May 2011 (UTC)
Hi all,
I came up with a way to add variance to the binomial distribution, for this purpose I consider the history of success compare to the expected value. Here is my development (I hope it is OK to have a link)
I would really like to know what you think, to me it looks very cool as I use expected value and sum of binomial series and I didn't see anything like it anywhere.
What do you say?
Ofermano (talk) 16:31, 28 June 2011 (UTC)
Would it be possible to write an introductory section that gives just a conceptual description of what the binomial distribution is about, before we enter the maths? Like tossing a coin, or drawing marbles from a box, and replacing the drawn marble each time (and mixing the box up again)? --JN466 02:18, 3 July 2011 (UTC)
Hi, isn't the standard deviation calculated as : sqrt((p(1 − p) n)) ? In the article it is written as: sqrt((p(1 − p)/n)) — Preceding unsigned comment added by 213.55.184.169 (talk) 06:31, 15 March 2012 (UTC)
Is it my imagination or are only the first and last probabilities for the biased coin correct? I have run that in SAS
data _null_ ; p = 0.3 ; do i = 0 to 6 ; prob= (p**i) * ((1-p)**(6-i)) ; put i= prob= ; end ; run ;
and I get
i=0 prob=0.117649 i=1 prob=0.050421 i=2 prob=0.021609 i=3 prob=0.009261 i=4 prob=0.003969 i=5 prob=0.001701 i=6 prob=0.000729
Docsteve.518 (talk) 21:46, 15 April 2013 (UTC)
Oh yes, thanks for that. That's what I get for trying to do it long hand.
The SAS functions exist for a reason!
data _null_ ; do i = 0 to 6 ; x = pmf('Binomial',i,.3,6) ; put i= x= ; end ; run ;
And yes, there's the sequence
i=0 x=0.117649 i=1 x=0.302526 i=2 x=0.324135 i=3 x=0.18522 i=4 x=0.059535 i=5 x=0.010206 i=6 x=0.000729
16:07, 16 April 2013 (UTC) — Preceding unsigned comment added by 72.43.218.26 (talk)
Firstly, I want I am wondering about the definition of the CDF
From my training (and looking at the graphs on THIS page), we should be defining F on the real numbers and writing < and not ≤, that is:
or indeed:
(I also switched to j in place of i since so many applications now assume i is the corresponding complex number.)
I had already given this formula in my MK wikipedia page and was wanting to add an iterative "computer graphing formula" and so checked to see if there was one on this page and became confused with the above formula.
BTW: Here is the iterative formula I was getting ready to add. Any suggestions here to make this clearer how to use?
So if you make a sequence of the probabilities values (easy), you can easily make this into a sequence of n+2 points and then draw segments as in the above graphs. Having worked this out a gazillion times for my kiddies, I finally wrote it down.
Lfahlberg (talk) 09:06, 22 November 2013 (UTC)
Please label the axes on the graphs, and state the allowable range for parameter n. Does n include zero? 71.139.165.140 (talk) 19:11, 14 December 2014 (UTC)
Here is a reference to use: http://www.le.ac.uk/users/dsgp1/COURSES/MATHSTAT/5binomgf.pdf
Tal Galili (talk) 17:04, 25 March 2015 (UTC)
Cheers, Tal Galili (talk) 22:28, 25 March 2015 (UTC)
title says it all, see for example Ching-Hui Chang et. al.: "A note on Improved Approximation of the Binomiual Distribution by the Skew-Normal distribution" the American Statistician. Kjetil B Halvorsen 13:34, 8 June 2015 (UTC) — Preceding unsigned comment added by Kjetil1001 (talk • contribs)
The proof of the mode doesn't define a_k. It's relatively clear that f(k) is meant, but that should be defined (or f used) — Preceding unsigned comment added by Andreas Mueller (talk • contribs) 00:21, 3 March 2016 (UTC)
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Is the "Spoilers!" in the conditional binomial proof section a joke? :) Nicolas Perrault (talk) 15:38, 1 May 2017 (UTC)
Well... it is funny, but it is recently-added vandalism so I removed it. Bosons (talk) 02:31, 2 May 2017 (UTC)
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I think at its current state, the lead section is very opaque for a layperson. It's so full of technical jargons that only a person already familiar with these statistical terms would care to understand what this concept means. I think the lead should be much more accessible. It should use very simple language or have very simple language definition for each opaque jargon-y term. It should also contain concrete examples, preferably historically relevant examples, associated with the concept, to provide a very basic, graspable understanding of the topic. Zaheen (talk) 08:05, 14 September 2017 (UTC)
The Wikipedia:Manual of Style/Lead section says: "The lead ... should ... establish context, explain why the topic is notable..." and "avoid difficult-to-understand terminology" "....with the goal of making the lead section accessible to as broad an audience as possible. Where uncommon terms are essential, they should be placed in context, linked and briefly defined. The subject should be placed in a context familiar to a normal reader." I don't think that is the case here at all. Zaheen (talk) 08:09, 14 September 2017 (UTC)
Isn't the mean not the sum, but the sum divided by n? Similar errors appear to be made with the mode, median, variance, etc. I think skewness, kurtosis, and entropy could also be affected, but I don't know. When we look at the definition of this distribution it is clearly between 0 and 1 for all allowed values of p, k, and n; and yet if the mean is truly given by p*n, then with p=0.5 and n=3 we get 1.5 which is outside all output values. StephenJohns00 (talk) 11:37, 31 August 2018 (UTC)
Currently the random variate section has this paragraph, which is kind of weird:
For one, you can just do n bernoulli trials. You would need an absurdly large n for this be inefficient. Second, we have closed form expressions for the pmf that always add up to one, so if it does not, you miscalculated. Third, using the cumulative probability distribution would probably work better than the pmf. — Preceding unsigned comment added by TheKing44 (talk • contribs) 05:49, 1 February 2019 (UTC)
Entropy is given as entropy =
Is it possible to give a reference for that formula given that it is not standard textbook knowledge. Also it is unclear, what the operator/function O of 1 over n stands for. Thanks. (Sorry for all breaches of etiquette.) — Preceding unsigned comment added by 92.217.250.44 (talk) 14:56, 12 July 2019 (UTC)
I'm new and do not know the etiquette. Hence I will not update the page. However, it just came to my attention that you do not have the formula for entropy of the binomial distribution. it is 2^-S= ((N-U)/N)^N * (U/(N-U))^U, where in your notation N=n and U=np. The unit of S is in bits. I hope this helps, Jens Adler Nielsen Jens Adler Nielsen (talk) 10:12, 25 August 2019 (UTC)
The current tail bound given for p=1/2, n/2>=k>=3n/8 fails for n=64, k=26. I believe, based on the Chernoff inequality immediately above it, the 16 in the exponent should be replaced with a 4. Also, I can't find a way to derive this apparent formula from anything in the provided source.
69.119.31.14 (talk) 14:28, 12 August 2019 (UTC)
Confirming the error. The closest formula in the source is in Proposition 7.3.2, page 46, and it provides the lower bound . — Preceding unsigned comment added by 176.150.242.62 (talk) 00:08, 6 November 2019 (UTC)
Hi all, it seems to me that what is described here as the Fisher information is actually the expected Fisher information, i.e. expectation of the Fisher information (where the expectation is taken with respect to the data)[1]
The actual Fisher information is:
For a derivation of the Fisher information, see example 2.10 of this book,[2] and for a derivation of how taking the expectation leads to see example 4.1 of the same book.
Should we change that in the page? Best
Ddreif (talk) 18:44, 14 November 2021 (UTC)
References
This article was the subject of a Wiki Education Foundation-supported course assignment, between 27 August 2021 and 19 December 2021. Further details are available on the course page. Peer reviewers: Ziyanggod, C.Hua Wang, Jiang1725.
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In some places the article uses Binomial (not at start of sentence) and in others binomial. Universemaster1 (talk) 14:01, 28 May 2022 (UTC)
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Something in Firefox? Found it: text color must be black. OveGjerlow (talk) 19:09, 20 September 2023 (UTC)