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This page and many others now give an error when attempting to view with the Dictionary.app on MacOS: "This page cannot be displayed in this application."
What's causing this, and how can it be fixed? — Preceding unsigned comment added by Canavalia ( talk • contribs) 14:35, 15 May 2023 (UTC)
Ohanian 01:26, 2005 Apr 8 (UTC)
Consider the expression
Fixing (n,p) it is the binomial distribution of i. Fixing (n,i) it is the (unnormalized) beta distribution of p. The article does not clarify this.
Bo Jacoby 10:02, 15 September 2005 (UTC)
I don't see what makes you think the article is not explicit about this point. You wrote this on Sepember 15th, when the version of September 6th was there, and that version is perfectly explicit about it. It says the density f(x) is defined on the interval [0, 1], and x where it appears in that formula is the same as what you're calling p above. How explicit can you get? Michael Hardy 23:06, 16 December 2005 (UTC)
... or did you mean it fails to clarify that the same expression defines both functions? OK, maybe you did mean that ... Michael Hardy 23:08, 16 December 2005 (UTC)
Should the paragraph on computing and from moments be moved from the main section to a new section on Parameter Estimation? When looking for MOM parameter estimates, I missed that paragraph, and other distributions have their own parameter estimation section (e.g. gamma).
I added a section for the four parameter estimation case using the method of moments. Elderton, (see section titled "History"), in his 1906 monograph "Frequency curves and correlation," fully discusses the four parameter case and contains the correct equations. These equations have been repeated by other authors in other books. It is curious that the classic reference by N.L.Johnson and S.Kotz, in their (1970) first edition of "Continuous Univariate Distributions Vol. 2" , Wiley, Chapter 21:Beta Distributions, page 44, equation (15) contains an important error. The support interval range is given as follows:
Where Johnson and Kotz use the identical nomenclature used by Elderton in 1906: for the sample kurtosis, for sample variance and "r" for
This is incorrect. The correct equation is given by Elderton in 1906:
This range can also be expressed in terms of the excess kurtosis or the kurtosis, as I have done in the article (but it will read very differently than in Johnson and Kotz, whether one uses the kurtosis or the excess kurtosis).
Dr. J. Rodal (
talk)
19:15, 7 September 2012 (UTC)
What is the entropy of a beta distribution? This document: http://www.rand.org/pubs/monograph_reports/MR1449/MR1449.appa.pdf has a formula (and it refers to "Cover and Thomas (1991)"). Can someone verify it? They say psi is the derivative of the gamma function, but usually psi represents the digamma function which is the derivative of the log of the gamma function. So I'm wondering if they have a typo. A5 01:13, 22 May 2006 (UTC)
Great question, great comments and great response. Yes, it is counter-intuitive that the differential entropy of the beta distribution is negative (for all values of α and β except α=β=1).
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
I was asked the following question: I've been trying to derive the expression for the differential entropy of the beta distribution. According to http://en.wikipedia.org/wiki/Talk%3ABeta_distribution#Entropy, you derived it using mathematica. I have just installed mathematica, but I cannot reproduce your result (my attempt is below). Please can you send me your 'code'. I tried
Expectation[-q*Log[q], q \[Distributed] BetaDistribution[a, b]]
and got
(a (HarmonicNumber[a] - HarmonicNumber[a + b]))/(a + b)
ANSWER: The following explicit integration gives the answer in terms of the PolyGamma functions
In[1]:= FullSimplify[ Integrate[-Expand[(((1 - x)^(-1 + beta) x^(-1 + alpha) ) ((-1 + beta) Log[1 - x] + (-1 + alpha) Log[x] - Log[Beta[alpha, beta]]))/Beta[alpha, beta]], {x, 0, 1}]]
Out[1]= ConditionalExpression[ Log[Beta[alpha, beta]] - (-1 + alpha) PolyGamma[0, alpha] - (-1 + beta) PolyGamma[0, beta] + (-2 + alpha + beta) PolyGamma[0, alpha + beta], Re[beta] > 0 && Re[alpha] > 0]
Dr. J. Rodal ( talk) 14:45, 6 November 2013 (UTC)
Notes:
1) Following Claude E. Shannon, the differential entropy is h = E[ - ln(q)] and not h = E[- q ln(q)] (with q \[Distributed] BetaDistribution[a, b] )
2) One has to guide Mathematica for this integration. There are issues as x approaches 0 and 1, and for alpha and beta approaching 0. The Mathematica (Versions 8 and 9) Expectation function is too general to use without conditioning the expression variables. Therefore it is best to use the Mathematica Integration function and perform an explicit integration term by term as:
instead of directly using the Mathematica (Versions 8 and 9) Expectation function as:
Dr. J. Rodal ( talk) 13:49, 7 November 2013 (UTC)
UPDATE (Nov 11, 2013): I sent the above to Wolfram Support and received the following answer (bold added for emphasis): "thank you for your feedback . Mathematica is not able to compute the Expectation (of the differential entropy of the beta distribution) even under the condition {alpha>0, beta>0} but is able to do so by the command you applied, will file a suggestion on this for Mathematica to have better support on computing the differential entropy of beta distribution (on future versions). Please let us know if you have any other comments or questions to our product and we will be glad to help you. support@wolfram.com"
Dr. J. Rodal ( talk) 00:47, 12 November 2013 (UTC)
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
It would be better to move some of the application section to the introduction to give people an idea of why this is usefull instead of its mathematical definition.
I agree with both comments, that discussing actual applications in the introduction would be appealing to a wide range of users, therefore:
I don't know the correct formula, but in the current formula, the summand does not depend on j. So, I assume it is wrong.
There are two forms for the Beta distribution. At present only the so-called 'Beta distribution of the first kind' is discussed. The Beta distribution of the second kind does not seem to be discussed in Wikipedia as I write. Rwb001 06:26, 30 September 2006 (UTC)
Why is there SO much blank, and therefore wasted, space on this page? —The preceding unsigned comment was added by Algebra man ( talk • contribs) 20:09, 8 December 2006 (UTC).
Is there a simple closed form for the median of the beta distribution?
(apart from quantile(half)?)
Paul A Bristow 13:33, 21 December 2006 (UTC) Paul A. Bristow
$$$$$$$$$$$$$$$$$$$$$ Very good question by Paul A Bristow and an excellent answer by Lovibond ( talk). I added a new section as follows [the exact solutions for α=3 and β=2 and vice-versa are lengthy expressions containing cubic and square roots, so I included it as follows to save space] ( Dr. J. Rodal ( talk) 19:12, 6 August 2012 (UTC)) :
(*NEW Section Begins
MEDIAN
The median of the beta distribution is the unique real number x for which the regularized incomplete beta function . There is no general closed-form expression for the median of the beta distribution for arbitrary values of α and β. Closed-form expressions for particular values of the parameters α and β follow:
A reasonable approximation, in the range α ≥ 1 and β ≥ 1, of the value of the median ν is given by the formula [6]
For α ≥ 1 and β ≥ 1, the relative error (the absolute error divided by the median) in this approximation is less than 4% and for both α ≥ 2 and β ≥ 2 it is less than 1%. The absolute error divided by the difference between the mean and the mode is similarly small:
NEW Section Ends*) $$$$$$$$$$$$$$$$$$$$$
References
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I dont'know their distribution? What is F1 in the Char Function? —The preceding unsigned comment was added by 148.235.65.243 ( talk) 18:40, 12 February 2007 (UTC).
I agree with this question, what is in the characteristic function. I can find nothing on it at Mathworld, nor in Peebles, nor in Papoulis, nor in Zwillinger, and that's all my references. Anybody know?--
Phays
22:14, 17 August 2007 (UTC)
Anybody still interested in the characteristic function of the beta distribution, the confluent hypergeometric function of the first kind , may also find section 9.2 of Gradshteyn,I. S. , and I. M. Ryzhik, Table of Integrals, Series, and Products, year=2000, Academic Press; 6th edition, isbn=78-0122947575, of benefit . Presently, this classic reference is not cited among the references in Wikipedia's article on
Confluent hypergeometric functions — Preceding
unsigned comment added by
Dr. J. Rodal (
talk •
contribs)
17:23, 8 August 2012 (UTC)
I added a new section on the
characteristic function of the beta distribution: the
confluent hypergeometric function (of the first kind) where I point out that in the symmetric case α = β it simplifies to a
Bessel function using
Kummer's second transformation as follows:
I also included accompanying plots, the real part (Re) of the characteristic function of the beta distribution displayed for symmetric (α = β) and skewed (α ≠ β) cases. Dr. J. Rodal ( talk) 11:39, 16 August 2012 (UTC)
Is anyone able to add a section how you would draw random samples from the beta distribution? Is there a direct method like a transform from uniform variates, or do you have to use rejection sampling? 195.157.136.194 08:50, 22 January 2007 (UTC)
148.235.65.243 18:45, 12 February 2007 (UTC)CRIstinaGH
For deviates from a Beta(a,b) random variable, where a and b strictly positive. Sample x1 from Gamma(a) and x2 from Gamma(b), the deviate is then x1/(x1+x2). You can find this in "Numerical Analysis for Statisticians" by Kenneth Lange 1999, chapter 20. The method I described is for a general Dirichlet (the N dimensional extension of the Beta) however Lange also gives a short description of rejection methods appropriate for Beta distributions with parameters > 1. 128.54.54.242 02:18, 13 March 2007 (UTC)
For another solution that involves rejection sampling, see Cheng "Generating Beta Variates with Nonintegral Shape Parameters", Communications of the ACM 1978. This is the algorithm cited by the R implementation of draws from a standard beta distribution. 209.6.93.33 ( talk) 01:38, 9 September 2010 (UTC)
This article gives the formula for a random number generator that produces results that fit within the beta distribution. I was wondering if that could be included somewhere in the article, as well. 76.252.15.202 ( talk) 05:08, 21 May 2008 (UTC)
Note that we need to assume independence of X and Y in order to conclude that X/(X+Y) generates a beta distribution when X and Y each follow a gamma distribution with the same scale parameter. For example X/(X+X) = 1/2 with probability one, which is not a proper Beta distribution. — Preceding unsigned comment added by Tthrall ( talk • contribs) 17:33, 24 November 2011 (UTC)
For the beta distribution:
has the form:
Has this form also a special name? —Preceding unsigned comment added by 84.83.33.64 ( talk) 09:13, 3 October 2008 (UTC)
Good point to bring up, and good answers as well. I added the β=1, α>1 case as the power function distribution among the "Shapes" cases. I also added the reverse case β>1, α=1 as the "reverse (mirror-image) power function distribution," This power function distribution is one of the few cases in which there is a closed-form solution for the Median,and therefore I also added it in that section. Dr. J. Rodal ( talk) 21:33, 8 August 2012 (UTC)
Is it worth mentioning that for large values of α + β the beta distribution converges towards a normal distribution?
in terms of a weighted-coin toss: if heads are worth 1 and tails are worth 0:
Dividing the sample-variance by the number of samples gives the expected variance of our sample's mean. —Preceding unsigned comment added by Sukisuki ( talk • contribs) 22:41, 10 April 2010 (UTC)
The interpretation of the domain of the beta function is defined in the first paragraph, but the interpretation of the range is not clearly defined. If this is a probability distribution, why can the range values go over 1.0? (I understand that the cumulative distribution reaches a range of 1.0 at a domain value of 1.0.) Please can some clarification be given? —Preceding unsigned comment added by 66.30.115.7 ( talk) 20:21, 24 October 2010 (UTC)
The present version has a supposed formula for the "kurtosis excess" that starts off with the expession for the 4th central moment only, so that part is wrong. But is the rest of the expression correct for either the excess kurtosis or the 4th moment? It doesn't coincide with any formula I can conveniently find. .... and it would be good provide explicit citations for this stuff. Melcombe ( talk) 13:03, 25 October 2010 (UTC)
Addressing the helpful points raised by User Melcombe:
The formula is the formula found in most books, and it can also be obtained, for example, from mathematical software like Mathematica. However, the formula : is also correct (one can verify this by expansion, or by using Mathematica for example) and it has the advantages to be more compact and to have the numerator in terms of the difference and the sum of α and β, the term in the numerator proportional to (α - β) vanishes for the (important) symmetric case α=β.
Dr. J. Rodal ( talk) 16:54, 4 August 2012 (UTC)
In contingency table analysis, and I'm sure in many other situations, the need to test the difference between two beta-distributed random variables arises. The posterior distribution of the difference is the convolution of the two original beta distributions. However I have seen unsubstantiated claims in a couple of places that the convolution of two beta distributions has no analytical form. Has it been proved that an analytical form doesn't exist, or is there just no known analytical form? I'm assuming you can still estimate the mean of the posterior as the difference of the means of the two beta distributions, and the variance of the posterior as the sum of the variances (if independent), or the sum of the variance plus twice the covariance (if the covariance is non-zero)? Is there a closed form for covariance between two beta distributions? I *think* that the 1st order Taylor series approximation of the covariance is given by as shown in Eq (4) of "An algorithm for generating positively correlated Beta-distributed random variables with known marginal distributions and a specified correlation", but I'm not sure if it's OK to drop the and variables.
It would be great if there could be some treatment of these topics in the main article. LukeH ( talk) 01:34, 27 October 2010 (UTC)
A useful parameterization of the Beta distribution is in terms of its mean and sample size. This is useful for Bayesian estimation... for example, one would typically place a uniform(0, 1) prior over the mean of a Beta distribution, and a vague prior over the sample size. This is much easier than specifying an equivalent prior over alpha and beta. I'm not up on wiki math code, so perhaps someone could add this information. The two parameter sets are related via: alpha = (mean) x (sample size), beta = (1 - mean) x (sample size). The text "Doing Bayesian Data Analysis" by Kruschke provides a reference for this (p. 83), I am sure there are others. 128.151.80.182 ( talk) 17:54, 16 February 2011 (UTC)
To grasp the meaning of the beta distribution, an intuitive interpretation and/or an illustratory example of application from everyday life would be very helpful for the layperson. --Typofier 08:23, 22 June 2011 (UTC) — Preceding unsigned comment added by Typofier ( talk • contribs)
The article currently defines the Dirichet distribution as "the multivariate generalization of the beta distribution." Certainly, there are other multivariate distributions, such as those for random matrices. Refer, for example, to C. G. Khatri, "On the mutual independence of certain statistics." Annals of Mathematical Statistics, 30 : 4 : 1258--1262 (1959). Accordingly, I have changed "the" to "a" in the quoted portion of the article. Lovibond ( talk) 03:54, 22 September 2011 (UTC)
Currently the Mean and the Variance based on Ln[X] are in the property box for the Beta Distribution:
I presume that these have been added because the Ln[X] transformation extends the distribution from a bounded [0,1] domain for X to a semi-infinite Ln[X] domain (for X approaching zero) ?.
QUESTION: Should we have these Ln[X] properties in the statistical property box rather than just in the text?
The same could be done for other statistical distributions on a bounded domain. However, I could not find such an addition in the property box for other distributions. I don't feel strongly about this one way or another, but it would be nice to arrive at some convention as to what properties should be included in the statistical property box for distributions in Wikipedia.
Also, the Ln[X] transformation naturally leads to a discussion of Johnson distributions, which I have not found presently in Wikipedia.
Thanks,
Dr. J. Rodal ( talk) 14:53, 11 August 2012 (UTC)
It's quite useful for statisticians to have logarithmic expectations in some kind of look up table, as statistical procedures (eg EM Variational Bayes) sometimes involve taking expectations of log-likelihoods or log-pdfs. That said, as you say most of the standard +ve valued continuous distributions don't have logarithmic expectations listed. Perhaps they should be added? There maybe less of a case to include logarithmic variances: is there a case for a separate article for logarithmic-beta distribution? What is a Johnson distribution? Wjastle ( talk) 19:54, 11 August 2012 (UTC)
Great response, Wjastle, you make very good points, and clearly stated. Johnson distributions are a family of distributions proposed by Johnson in 1949 (Biometrika Vol 36 p.148) that are based on the transformation of a standard normal variate. He proposed logarithmic and hyperbolic sine transformations with varying number of parameters. One of them, the Johnson SB transformation overlaps the beta distribution in the Pearson (skewness^2,kurtosis) plane, and actually covers a greater area than the beta distribution. Another, the SU distribution is able to cover regions of the (skewness^2,kurtosis) that are not covered by any of the Pearson distributions ( http://en.wikipedia.org/wiki/Pearson_distribution ). I just made a cursory search with Google (only looking at the first few entries) and I did not find any comprehensive article, but the following one has an amusing description:
http://www.financialwebring.org/gummy-stuff/johnson.htm
Thanks Dr. J. Rodal ( talk) 22:26, 11 August 2012 (UTC)
I added a new section titled "Kurtosis bounded by the square of the skewness," which contains the following equation:
<<
The region occupied by the beta distribution is bounded by the following two lines in the (skewness2,kurtosis) plane, or the (skewness2,excess kurtosis) plane:
>>
I am aware that the (otherwise excellent) reference "Gupta (Editor), Arjun K. (2004). Handbook of Beta Distribution and Its Applications. CRC Press. pp. 42. ISBN 978-0824753962." quoted in the rest of this Wikipedia article instead has this equation on page 42 (in section VII of the chapter "Mathematical properties of the Beta distribution" by Gupta and Nadarajah):
Gupta et.al quotes Karian (1996) as the source of this equation. The lower bound for this equation is correct: it is the "impossible region" previously found by K. Pearson. However, the upper bound equation in Gupta:
is incorrect. The correct equation is:
as correctly found by K.Pearson practically a century ago (one can also verify this by numerical examples, as I included in the Wikipedia article). Dr. J. Rodal ( talk) 22:20, 12 August 2012 (UTC)
I added a new section titled "Mean absolute deviation around the mean". A few notes:
Dr. J. Rodal ( talk) 15:36, 23 August 2012 (UTC)
I wrote a section on the Fisher information matrix for the beta distribution. I framed its derivation in terms of the log likelihood function (as done for example by E.T.Jaynes in "Probability theory, the logic of science", A.W.F. Edwards in "Likelihood", and several others) instead of the probability density function conditional on the value of a parameter as done in the Wikipedia article on Fisher Information, to emphasize its main role in parameter estimation. These two ways to frame it are equivalent and whether to chose one or another is a matter of preference and the context in which one is writing. For the four parameter case I quote the recent article by Aryal and Nadarajah because it may be more readily available to an Internet audience, since its ".pdf" is presently freely available through the Internet. This article by Aryal and Nadarajah contains an error for the Fisher information component: the component given by the log likelihood function's second order derivative with respect to the minimum "a" (or "c" in Aryal and Nadarajah's nomenclature). The expression in their article is incorrect in several ways, and it appears incorrectly twice (first as the second order partial derivative of the likelihood function and then again as the expected value for the Fisher information component). I have corrected their error and placed the correct equation in the Wikipedia article. One can verify this by obtaining the correct equation by differentiation of the log likelihood function and carrying out the Integration for the expected value, one can readily see that this Fisher information component is incorrect in Aryal and Nadarajah from symmetry arguments, since the Fisher information terms given by the second order differentiation with respect to the minimum "a" and the Fisher information component given by the second order differentiation with respect to the maximum "c" (or "d" in Aryal and Nadarajah's nomenclature) should be symmetric. It is curious that Aryal and Nadarajah do not use the trigamma function in their expression for the first three Fisher information matrix components, and instead express it in terms of the (more lengthy expressions for the) derivatives of the gamma function. Dr. J. Rodal ( talk) 18:11, 13 September 2012 (UTC)
Dr. J. Rodall, In this section, is your expression for the log likelihood correct? You have: log likelihood(p|H) = H*log(p) - (1-H)*log(1-p) Except, shouldn't the likelihood be: log likelihood(p|H) = log[Pr(H|p)] = log[p^H * (1-p)^(1-H)] = log[p^H] + log[(1-p)^(1-H)] = H*log(p) + (1-H)*log(1-p) Sorry if I am missing something. Thanks for the help! — Preceding unsigned comment added by 18.111.7.78 ( talk) 22:57, 17 October 2012 (UTC)
Dear User:18.111.7.78, thanks for catching this in the section "Jeffreys' prior probability ( Beta(1/2,1/2) for a Bernoulli or for a binomial distribution )", you are correct: the (-) should have been a (+) and it should have read log likelihood(p|H)= H*log(p) + (1-H)*log(1-p). I made this correction in the text as well.( Dr. J. Rodal ( talk) 02:56, 19 October 2012 (UTC)
"It has also been" is used 5 times in a row in the lead. Boooring.... :-) I guess the intro is too loong and descriptive and most of those uses (all of them?) could go into a (new and first) section of their own. - Nabla ( talk) 14:02, 18 September 2012 (UTC)
Discounting lists, this article is currently one of the longest on Wikipedia. The recent additions by User:Dr. J. Rodal have been fantastic but it is extremely disorienting to attempt to read through the article in its current state. I thought I would create a section to discuss ways in which it can be pruned. Some initial ideas:
I don't have any experience with writing articles on probability distributions on Wikipedia (but plenty of experience of reading them) so I thought I'd see what others think. -- Iae ( talk) 12:31, 10 October 2012 (UTC)
I tend to be suspicious of complaints of this kind ("This article is too (1) technical; (2) long; (3) unimportant; (4) whatever."). I can begin to why someone might think this one is too long. Before opining further, I'll have to look at it further. Michael Hardy ( talk) 17:45, 10 October 2012 (UTC)
revision 527168631 by 99.241.86.114 placed the Mean and Sample Size Parametrization in the introductory paragraph. This can be justified by the fact that this parametrization is discussed throughout the text in different sections. However, other more (historically) important parametrizations (for example the four parameter parametrization) are also discussed throughout the text. Is there a reason why (only) the Mean and Sample Size should be discussed in the introductory paragraph ? Should all parametrizations be discussed in the introductory paragraph? Isn't it adequate to discuss the parametrizations under their own section? Alternatively, should the parametrization section be moved near the top of the article? Dr. J. Rodal ( talk) 17:56, 9 December 2012 (UTC)
Ferrari and Cribari-Neto (2004) propose a Beta regression algorithm. This approach is not very popular (doesn't have a wikipedia page) but I think it's worth mentioning it here. — Preceding unsigned comment added by 134.191.232.71 ( talk) 08:29, 19 February 2013 (UTC)
For anybody interested in "Beta Regression", see: Ferrari and Cribari-Neto and Cribari-Neto and Zeileis . Dr. J. Rodal ( talk) 14:18, 19 February 2013 (UTC)
Unless you already have enough grounding in mathematics to already know what the Beta Distribution is this article never actually explains it. The first sentence (which is supposed to be accessible to lay people) is just a jargon filled way of saying "its an equation with some variables in it". — Preceding unsigned comment added by 167.206.48.220 ( talk) 21:02, 20 February 2013 (UTC)
From one point of view, the article is great as a quite detailed description of the beta distribution and its properties. However, considering wikipedia an encyclopedia, I have serious doubts about the article's usefulness. If I need to find something quite general, I skim through encyclopedia (e.g. wikipedia). For details, I never do this, because it's not the purpose of any encyclopedia to be exhaustive in any entry. I believe that focused books/papers are appropriate for that. In this respect, the beta distribution article on wikipedia fails to deliver the "encyclopedic message". I honestly respect the great effort of the author and I'm not sure about any wiki-consistent modification. Maybe to move the detailed content to another article like "Beta distribution (details)"? — Preceding unsigned comment added by 77.78.117.1 ( talk) 09:23, 7 September 2013 (UTC)
This section is unnecessarily wordy, and ends up repeating a lot of the same information. While I appreciate that it's worth pointing out the difference between the Jeffreys, Bayes and Haldane priors is it really necessary to specify their posterior distributions individually? And after that there is a huge block of text which deals with the s=0 and n=0 cases in such a way that there is lots of text that says nothing. Most of this could go.
118.208.48.83 ( talk) 20:04, 24 October 2013 (UTC)
At User_talk:Jimbo_Wales#Search_recent_changes.3F, someone has pretended that page Beta distribution is owned by a terrific dragon that delete any contribution that tries to make the article clearer and shorter (see https://en.wikipedia.org/?title=User_talk%3AJimbo_Wales&type=revision&diff=710796589&oldid=710789244). Let us try if this assertion contains any part of truth. Pldx1 ( talk) 17:32, 19 March 2016 (UTC)
The inbox says x ∈ (0,1) while the first paragraph says it is [0,1]. which is it? 107.133.211.254 ( talk) 12:05, 6 August 2016 (UTC)
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The equation in the Probability Density Function section in the beginning of the article uses the parameter "u" in the denominator but fails to mention what "u" is. Clarification would be appreciated. — Preceding unsigned comment added by Pwvirgo ( talk • contribs) 15:23, 29 July 2017 (UTC)
I believe that there is an error in the following formula:
It should be the covariance between ln(X) and ln(1-X) instead of between ln(X) and 1-X:
This would be consistent with the formula in Section 3.3.1 for the 2 parameter case. However, I am not entirely sure, so perhaps someone else can also have a look at this.
2A00:E040:C0C:4501:0:0:0:167 ( talk) 10:25, 24 January 2018 (UTC)
The first sentence suggests that the distribution is a function of some random variable, which surely is not correct. I propose to remove "appear as exponents of the random variable and". — Preceding unsigned comment added by Prof. Globi ( talk • contribs) 22:31, 20 March 2019 (UTC)
The following two sections should be merged:
Tal Galili ( talk) 11:40, 30 November 2019 (UTC)
No mistake. Sorry. I don't know how to remove this comment.
The article states that in a particular limit, the Beta distribution approaches the Gamma distribution. Is there a source or derivation for this? Also, should it also mention approaching the Delta distribution in another limit? Cesiumfrog ( talk) 00:17, 4 August 2020 (UTC)
I tried to translate the article into Czech with the Wikipedia built-in translator, but I was forced to stop after a few sentences. The article is so incredibly long and equation/graphics heavy that it breaks the tool. Can please somebody knowledgeable divide the article into a set of shorter ones? There is a lot of material here which is of interest only for very specialized experts. Jan Spousta ( talk) 11:24, 22 December 2020 (UTC)
Hey :) I think the article is very rich, but that's the expected type of material about such a central distribution. It's not much different then the normal distribution, binomial distribution l, etc. Tal Galili ( talk) 18:50, 22 December 2020 (UTC)
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This page and many others now give an error when attempting to view with the Dictionary.app on MacOS: "This page cannot be displayed in this application."
What's causing this, and how can it be fixed? — Preceding unsigned comment added by Canavalia ( talk • contribs) 14:35, 15 May 2023 (UTC)
Ohanian 01:26, 2005 Apr 8 (UTC)
Consider the expression
Fixing (n,p) it is the binomial distribution of i. Fixing (n,i) it is the (unnormalized) beta distribution of p. The article does not clarify this.
Bo Jacoby 10:02, 15 September 2005 (UTC)
I don't see what makes you think the article is not explicit about this point. You wrote this on Sepember 15th, when the version of September 6th was there, and that version is perfectly explicit about it. It says the density f(x) is defined on the interval [0, 1], and x where it appears in that formula is the same as what you're calling p above. How explicit can you get? Michael Hardy 23:06, 16 December 2005 (UTC)
... or did you mean it fails to clarify that the same expression defines both functions? OK, maybe you did mean that ... Michael Hardy 23:08, 16 December 2005 (UTC)
Should the paragraph on computing and from moments be moved from the main section to a new section on Parameter Estimation? When looking for MOM parameter estimates, I missed that paragraph, and other distributions have their own parameter estimation section (e.g. gamma).
I added a section for the four parameter estimation case using the method of moments. Elderton, (see section titled "History"), in his 1906 monograph "Frequency curves and correlation," fully discusses the four parameter case and contains the correct equations. These equations have been repeated by other authors in other books. It is curious that the classic reference by N.L.Johnson and S.Kotz, in their (1970) first edition of "Continuous Univariate Distributions Vol. 2" , Wiley, Chapter 21:Beta Distributions, page 44, equation (15) contains an important error. The support interval range is given as follows:
Where Johnson and Kotz use the identical nomenclature used by Elderton in 1906: for the sample kurtosis, for sample variance and "r" for
This is incorrect. The correct equation is given by Elderton in 1906:
This range can also be expressed in terms of the excess kurtosis or the kurtosis, as I have done in the article (but it will read very differently than in Johnson and Kotz, whether one uses the kurtosis or the excess kurtosis).
Dr. J. Rodal (
talk)
19:15, 7 September 2012 (UTC)
What is the entropy of a beta distribution? This document: http://www.rand.org/pubs/monograph_reports/MR1449/MR1449.appa.pdf has a formula (and it refers to "Cover and Thomas (1991)"). Can someone verify it? They say psi is the derivative of the gamma function, but usually psi represents the digamma function which is the derivative of the log of the gamma function. So I'm wondering if they have a typo. A5 01:13, 22 May 2006 (UTC)
Great question, great comments and great response. Yes, it is counter-intuitive that the differential entropy of the beta distribution is negative (for all values of α and β except α=β=1).
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
I was asked the following question: I've been trying to derive the expression for the differential entropy of the beta distribution. According to http://en.wikipedia.org/wiki/Talk%3ABeta_distribution#Entropy, you derived it using mathematica. I have just installed mathematica, but I cannot reproduce your result (my attempt is below). Please can you send me your 'code'. I tried
Expectation[-q*Log[q], q \[Distributed] BetaDistribution[a, b]]
and got
(a (HarmonicNumber[a] - HarmonicNumber[a + b]))/(a + b)
ANSWER: The following explicit integration gives the answer in terms of the PolyGamma functions
In[1]:= FullSimplify[ Integrate[-Expand[(((1 - x)^(-1 + beta) x^(-1 + alpha) ) ((-1 + beta) Log[1 - x] + (-1 + alpha) Log[x] - Log[Beta[alpha, beta]]))/Beta[alpha, beta]], {x, 0, 1}]]
Out[1]= ConditionalExpression[ Log[Beta[alpha, beta]] - (-1 + alpha) PolyGamma[0, alpha] - (-1 + beta) PolyGamma[0, beta] + (-2 + alpha + beta) PolyGamma[0, alpha + beta], Re[beta] > 0 && Re[alpha] > 0]
Dr. J. Rodal ( talk) 14:45, 6 November 2013 (UTC)
Notes:
1) Following Claude E. Shannon, the differential entropy is h = E[ - ln(q)] and not h = E[- q ln(q)] (with q \[Distributed] BetaDistribution[a, b] )
2) One has to guide Mathematica for this integration. There are issues as x approaches 0 and 1, and for alpha and beta approaching 0. The Mathematica (Versions 8 and 9) Expectation function is too general to use without conditioning the expression variables. Therefore it is best to use the Mathematica Integration function and perform an explicit integration term by term as:
instead of directly using the Mathematica (Versions 8 and 9) Expectation function as:
Dr. J. Rodal ( talk) 13:49, 7 November 2013 (UTC)
UPDATE (Nov 11, 2013): I sent the above to Wolfram Support and received the following answer (bold added for emphasis): "thank you for your feedback . Mathematica is not able to compute the Expectation (of the differential entropy of the beta distribution) even under the condition {alpha>0, beta>0} but is able to do so by the command you applied, will file a suggestion on this for Mathematica to have better support on computing the differential entropy of beta distribution (on future versions). Please let us know if you have any other comments or questions to our product and we will be glad to help you. support@wolfram.com"
Dr. J. Rodal ( talk) 00:47, 12 November 2013 (UTC)
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
It would be better to move some of the application section to the introduction to give people an idea of why this is usefull instead of its mathematical definition.
I agree with both comments, that discussing actual applications in the introduction would be appealing to a wide range of users, therefore:
I don't know the correct formula, but in the current formula, the summand does not depend on j. So, I assume it is wrong.
There are two forms for the Beta distribution. At present only the so-called 'Beta distribution of the first kind' is discussed. The Beta distribution of the second kind does not seem to be discussed in Wikipedia as I write. Rwb001 06:26, 30 September 2006 (UTC)
Why is there SO much blank, and therefore wasted, space on this page? —The preceding unsigned comment was added by Algebra man ( talk • contribs) 20:09, 8 December 2006 (UTC).
Is there a simple closed form for the median of the beta distribution?
(apart from quantile(half)?)
Paul A Bristow 13:33, 21 December 2006 (UTC) Paul A. Bristow
$$$$$$$$$$$$$$$$$$$$$ Very good question by Paul A Bristow and an excellent answer by Lovibond ( talk). I added a new section as follows [the exact solutions for α=3 and β=2 and vice-versa are lengthy expressions containing cubic and square roots, so I included it as follows to save space] ( Dr. J. Rodal ( talk) 19:12, 6 August 2012 (UTC)) :
(*NEW Section Begins
MEDIAN
The median of the beta distribution is the unique real number x for which the regularized incomplete beta function . There is no general closed-form expression for the median of the beta distribution for arbitrary values of α and β. Closed-form expressions for particular values of the parameters α and β follow:
A reasonable approximation, in the range α ≥ 1 and β ≥ 1, of the value of the median ν is given by the formula [6]
For α ≥ 1 and β ≥ 1, the relative error (the absolute error divided by the median) in this approximation is less than 4% and for both α ≥ 2 and β ≥ 2 it is less than 1%. The absolute error divided by the difference between the mean and the mode is similarly small:
NEW Section Ends*) $$$$$$$$$$$$$$$$$$$$$
References
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I dont'know their distribution? What is F1 in the Char Function? —The preceding unsigned comment was added by 148.235.65.243 ( talk) 18:40, 12 February 2007 (UTC).
I agree with this question, what is in the characteristic function. I can find nothing on it at Mathworld, nor in Peebles, nor in Papoulis, nor in Zwillinger, and that's all my references. Anybody know?--
Phays
22:14, 17 August 2007 (UTC)
Anybody still interested in the characteristic function of the beta distribution, the confluent hypergeometric function of the first kind , may also find section 9.2 of Gradshteyn,I. S. , and I. M. Ryzhik, Table of Integrals, Series, and Products, year=2000, Academic Press; 6th edition, isbn=78-0122947575, of benefit . Presently, this classic reference is not cited among the references in Wikipedia's article on
Confluent hypergeometric functions — Preceding
unsigned comment added by
Dr. J. Rodal (
talk •
contribs)
17:23, 8 August 2012 (UTC)
I added a new section on the
characteristic function of the beta distribution: the
confluent hypergeometric function (of the first kind) where I point out that in the symmetric case α = β it simplifies to a
Bessel function using
Kummer's second transformation as follows:
I also included accompanying plots, the real part (Re) of the characteristic function of the beta distribution displayed for symmetric (α = β) and skewed (α ≠ β) cases. Dr. J. Rodal ( talk) 11:39, 16 August 2012 (UTC)
Is anyone able to add a section how you would draw random samples from the beta distribution? Is there a direct method like a transform from uniform variates, or do you have to use rejection sampling? 195.157.136.194 08:50, 22 January 2007 (UTC)
148.235.65.243 18:45, 12 February 2007 (UTC)CRIstinaGH
For deviates from a Beta(a,b) random variable, where a and b strictly positive. Sample x1 from Gamma(a) and x2 from Gamma(b), the deviate is then x1/(x1+x2). You can find this in "Numerical Analysis for Statisticians" by Kenneth Lange 1999, chapter 20. The method I described is for a general Dirichlet (the N dimensional extension of the Beta) however Lange also gives a short description of rejection methods appropriate for Beta distributions with parameters > 1. 128.54.54.242 02:18, 13 March 2007 (UTC)
For another solution that involves rejection sampling, see Cheng "Generating Beta Variates with Nonintegral Shape Parameters", Communications of the ACM 1978. This is the algorithm cited by the R implementation of draws from a standard beta distribution. 209.6.93.33 ( talk) 01:38, 9 September 2010 (UTC)
This article gives the formula for a random number generator that produces results that fit within the beta distribution. I was wondering if that could be included somewhere in the article, as well. 76.252.15.202 ( talk) 05:08, 21 May 2008 (UTC)
Note that we need to assume independence of X and Y in order to conclude that X/(X+Y) generates a beta distribution when X and Y each follow a gamma distribution with the same scale parameter. For example X/(X+X) = 1/2 with probability one, which is not a proper Beta distribution. — Preceding unsigned comment added by Tthrall ( talk • contribs) 17:33, 24 November 2011 (UTC)
For the beta distribution:
has the form:
Has this form also a special name? —Preceding unsigned comment added by 84.83.33.64 ( talk) 09:13, 3 October 2008 (UTC)
Good point to bring up, and good answers as well. I added the β=1, α>1 case as the power function distribution among the "Shapes" cases. I also added the reverse case β>1, α=1 as the "reverse (mirror-image) power function distribution," This power function distribution is one of the few cases in which there is a closed-form solution for the Median,and therefore I also added it in that section. Dr. J. Rodal ( talk) 21:33, 8 August 2012 (UTC)
Is it worth mentioning that for large values of α + β the beta distribution converges towards a normal distribution?
in terms of a weighted-coin toss: if heads are worth 1 and tails are worth 0:
Dividing the sample-variance by the number of samples gives the expected variance of our sample's mean. —Preceding unsigned comment added by Sukisuki ( talk • contribs) 22:41, 10 April 2010 (UTC)
The interpretation of the domain of the beta function is defined in the first paragraph, but the interpretation of the range is not clearly defined. If this is a probability distribution, why can the range values go over 1.0? (I understand that the cumulative distribution reaches a range of 1.0 at a domain value of 1.0.) Please can some clarification be given? —Preceding unsigned comment added by 66.30.115.7 ( talk) 20:21, 24 October 2010 (UTC)
The present version has a supposed formula for the "kurtosis excess" that starts off with the expession for the 4th central moment only, so that part is wrong. But is the rest of the expression correct for either the excess kurtosis or the 4th moment? It doesn't coincide with any formula I can conveniently find. .... and it would be good provide explicit citations for this stuff. Melcombe ( talk) 13:03, 25 October 2010 (UTC)
Addressing the helpful points raised by User Melcombe:
The formula is the formula found in most books, and it can also be obtained, for example, from mathematical software like Mathematica. However, the formula : is also correct (one can verify this by expansion, or by using Mathematica for example) and it has the advantages to be more compact and to have the numerator in terms of the difference and the sum of α and β, the term in the numerator proportional to (α - β) vanishes for the (important) symmetric case α=β.
Dr. J. Rodal ( talk) 16:54, 4 August 2012 (UTC)
In contingency table analysis, and I'm sure in many other situations, the need to test the difference between two beta-distributed random variables arises. The posterior distribution of the difference is the convolution of the two original beta distributions. However I have seen unsubstantiated claims in a couple of places that the convolution of two beta distributions has no analytical form. Has it been proved that an analytical form doesn't exist, or is there just no known analytical form? I'm assuming you can still estimate the mean of the posterior as the difference of the means of the two beta distributions, and the variance of the posterior as the sum of the variances (if independent), or the sum of the variance plus twice the covariance (if the covariance is non-zero)? Is there a closed form for covariance between two beta distributions? I *think* that the 1st order Taylor series approximation of the covariance is given by as shown in Eq (4) of "An algorithm for generating positively correlated Beta-distributed random variables with known marginal distributions and a specified correlation", but I'm not sure if it's OK to drop the and variables.
It would be great if there could be some treatment of these topics in the main article. LukeH ( talk) 01:34, 27 October 2010 (UTC)
A useful parameterization of the Beta distribution is in terms of its mean and sample size. This is useful for Bayesian estimation... for example, one would typically place a uniform(0, 1) prior over the mean of a Beta distribution, and a vague prior over the sample size. This is much easier than specifying an equivalent prior over alpha and beta. I'm not up on wiki math code, so perhaps someone could add this information. The two parameter sets are related via: alpha = (mean) x (sample size), beta = (1 - mean) x (sample size). The text "Doing Bayesian Data Analysis" by Kruschke provides a reference for this (p. 83), I am sure there are others. 128.151.80.182 ( talk) 17:54, 16 February 2011 (UTC)
To grasp the meaning of the beta distribution, an intuitive interpretation and/or an illustratory example of application from everyday life would be very helpful for the layperson. --Typofier 08:23, 22 June 2011 (UTC) — Preceding unsigned comment added by Typofier ( talk • contribs)
The article currently defines the Dirichet distribution as "the multivariate generalization of the beta distribution." Certainly, there are other multivariate distributions, such as those for random matrices. Refer, for example, to C. G. Khatri, "On the mutual independence of certain statistics." Annals of Mathematical Statistics, 30 : 4 : 1258--1262 (1959). Accordingly, I have changed "the" to "a" in the quoted portion of the article. Lovibond ( talk) 03:54, 22 September 2011 (UTC)
Currently the Mean and the Variance based on Ln[X] are in the property box for the Beta Distribution:
I presume that these have been added because the Ln[X] transformation extends the distribution from a bounded [0,1] domain for X to a semi-infinite Ln[X] domain (for X approaching zero) ?.
QUESTION: Should we have these Ln[X] properties in the statistical property box rather than just in the text?
The same could be done for other statistical distributions on a bounded domain. However, I could not find such an addition in the property box for other distributions. I don't feel strongly about this one way or another, but it would be nice to arrive at some convention as to what properties should be included in the statistical property box for distributions in Wikipedia.
Also, the Ln[X] transformation naturally leads to a discussion of Johnson distributions, which I have not found presently in Wikipedia.
Thanks,
Dr. J. Rodal ( talk) 14:53, 11 August 2012 (UTC)
It's quite useful for statisticians to have logarithmic expectations in some kind of look up table, as statistical procedures (eg EM Variational Bayes) sometimes involve taking expectations of log-likelihoods or log-pdfs. That said, as you say most of the standard +ve valued continuous distributions don't have logarithmic expectations listed. Perhaps they should be added? There maybe less of a case to include logarithmic variances: is there a case for a separate article for logarithmic-beta distribution? What is a Johnson distribution? Wjastle ( talk) 19:54, 11 August 2012 (UTC)
Great response, Wjastle, you make very good points, and clearly stated. Johnson distributions are a family of distributions proposed by Johnson in 1949 (Biometrika Vol 36 p.148) that are based on the transformation of a standard normal variate. He proposed logarithmic and hyperbolic sine transformations with varying number of parameters. One of them, the Johnson SB transformation overlaps the beta distribution in the Pearson (skewness^2,kurtosis) plane, and actually covers a greater area than the beta distribution. Another, the SU distribution is able to cover regions of the (skewness^2,kurtosis) that are not covered by any of the Pearson distributions ( http://en.wikipedia.org/wiki/Pearson_distribution ). I just made a cursory search with Google (only looking at the first few entries) and I did not find any comprehensive article, but the following one has an amusing description:
http://www.financialwebring.org/gummy-stuff/johnson.htm
Thanks Dr. J. Rodal ( talk) 22:26, 11 August 2012 (UTC)
I added a new section titled "Kurtosis bounded by the square of the skewness," which contains the following equation:
<<
The region occupied by the beta distribution is bounded by the following two lines in the (skewness2,kurtosis) plane, or the (skewness2,excess kurtosis) plane:
>>
I am aware that the (otherwise excellent) reference "Gupta (Editor), Arjun K. (2004). Handbook of Beta Distribution and Its Applications. CRC Press. pp. 42. ISBN 978-0824753962." quoted in the rest of this Wikipedia article instead has this equation on page 42 (in section VII of the chapter "Mathematical properties of the Beta distribution" by Gupta and Nadarajah):
Gupta et.al quotes Karian (1996) as the source of this equation. The lower bound for this equation is correct: it is the "impossible region" previously found by K. Pearson. However, the upper bound equation in Gupta:
is incorrect. The correct equation is:
as correctly found by K.Pearson practically a century ago (one can also verify this by numerical examples, as I included in the Wikipedia article). Dr. J. Rodal ( talk) 22:20, 12 August 2012 (UTC)
I added a new section titled "Mean absolute deviation around the mean". A few notes:
Dr. J. Rodal ( talk) 15:36, 23 August 2012 (UTC)
I wrote a section on the Fisher information matrix for the beta distribution. I framed its derivation in terms of the log likelihood function (as done for example by E.T.Jaynes in "Probability theory, the logic of science", A.W.F. Edwards in "Likelihood", and several others) instead of the probability density function conditional on the value of a parameter as done in the Wikipedia article on Fisher Information, to emphasize its main role in parameter estimation. These two ways to frame it are equivalent and whether to chose one or another is a matter of preference and the context in which one is writing. For the four parameter case I quote the recent article by Aryal and Nadarajah because it may be more readily available to an Internet audience, since its ".pdf" is presently freely available through the Internet. This article by Aryal and Nadarajah contains an error for the Fisher information component: the component given by the log likelihood function's second order derivative with respect to the minimum "a" (or "c" in Aryal and Nadarajah's nomenclature). The expression in their article is incorrect in several ways, and it appears incorrectly twice (first as the second order partial derivative of the likelihood function and then again as the expected value for the Fisher information component). I have corrected their error and placed the correct equation in the Wikipedia article. One can verify this by obtaining the correct equation by differentiation of the log likelihood function and carrying out the Integration for the expected value, one can readily see that this Fisher information component is incorrect in Aryal and Nadarajah from symmetry arguments, since the Fisher information terms given by the second order differentiation with respect to the minimum "a" and the Fisher information component given by the second order differentiation with respect to the maximum "c" (or "d" in Aryal and Nadarajah's nomenclature) should be symmetric. It is curious that Aryal and Nadarajah do not use the trigamma function in their expression for the first three Fisher information matrix components, and instead express it in terms of the (more lengthy expressions for the) derivatives of the gamma function. Dr. J. Rodal ( talk) 18:11, 13 September 2012 (UTC)
Dr. J. Rodall, In this section, is your expression for the log likelihood correct? You have: log likelihood(p|H) = H*log(p) - (1-H)*log(1-p) Except, shouldn't the likelihood be: log likelihood(p|H) = log[Pr(H|p)] = log[p^H * (1-p)^(1-H)] = log[p^H] + log[(1-p)^(1-H)] = H*log(p) + (1-H)*log(1-p) Sorry if I am missing something. Thanks for the help! — Preceding unsigned comment added by 18.111.7.78 ( talk) 22:57, 17 October 2012 (UTC)
Dear User:18.111.7.78, thanks for catching this in the section "Jeffreys' prior probability ( Beta(1/2,1/2) for a Bernoulli or for a binomial distribution )", you are correct: the (-) should have been a (+) and it should have read log likelihood(p|H)= H*log(p) + (1-H)*log(1-p). I made this correction in the text as well.( Dr. J. Rodal ( talk) 02:56, 19 October 2012 (UTC)
"It has also been" is used 5 times in a row in the lead. Boooring.... :-) I guess the intro is too loong and descriptive and most of those uses (all of them?) could go into a (new and first) section of their own. - Nabla ( talk) 14:02, 18 September 2012 (UTC)
Discounting lists, this article is currently one of the longest on Wikipedia. The recent additions by User:Dr. J. Rodal have been fantastic but it is extremely disorienting to attempt to read through the article in its current state. I thought I would create a section to discuss ways in which it can be pruned. Some initial ideas:
I don't have any experience with writing articles on probability distributions on Wikipedia (but plenty of experience of reading them) so I thought I'd see what others think. -- Iae ( talk) 12:31, 10 October 2012 (UTC)
I tend to be suspicious of complaints of this kind ("This article is too (1) technical; (2) long; (3) unimportant; (4) whatever."). I can begin to why someone might think this one is too long. Before opining further, I'll have to look at it further. Michael Hardy ( talk) 17:45, 10 October 2012 (UTC)
revision 527168631 by 99.241.86.114 placed the Mean and Sample Size Parametrization in the introductory paragraph. This can be justified by the fact that this parametrization is discussed throughout the text in different sections. However, other more (historically) important parametrizations (for example the four parameter parametrization) are also discussed throughout the text. Is there a reason why (only) the Mean and Sample Size should be discussed in the introductory paragraph ? Should all parametrizations be discussed in the introductory paragraph? Isn't it adequate to discuss the parametrizations under their own section? Alternatively, should the parametrization section be moved near the top of the article? Dr. J. Rodal ( talk) 17:56, 9 December 2012 (UTC)
Ferrari and Cribari-Neto (2004) propose a Beta regression algorithm. This approach is not very popular (doesn't have a wikipedia page) but I think it's worth mentioning it here. — Preceding unsigned comment added by 134.191.232.71 ( talk) 08:29, 19 February 2013 (UTC)
For anybody interested in "Beta Regression", see: Ferrari and Cribari-Neto and Cribari-Neto and Zeileis . Dr. J. Rodal ( talk) 14:18, 19 February 2013 (UTC)
Unless you already have enough grounding in mathematics to already know what the Beta Distribution is this article never actually explains it. The first sentence (which is supposed to be accessible to lay people) is just a jargon filled way of saying "its an equation with some variables in it". — Preceding unsigned comment added by 167.206.48.220 ( talk) 21:02, 20 February 2013 (UTC)
From one point of view, the article is great as a quite detailed description of the beta distribution and its properties. However, considering wikipedia an encyclopedia, I have serious doubts about the article's usefulness. If I need to find something quite general, I skim through encyclopedia (e.g. wikipedia). For details, I never do this, because it's not the purpose of any encyclopedia to be exhaustive in any entry. I believe that focused books/papers are appropriate for that. In this respect, the beta distribution article on wikipedia fails to deliver the "encyclopedic message". I honestly respect the great effort of the author and I'm not sure about any wiki-consistent modification. Maybe to move the detailed content to another article like "Beta distribution (details)"? — Preceding unsigned comment added by 77.78.117.1 ( talk) 09:23, 7 September 2013 (UTC)
This section is unnecessarily wordy, and ends up repeating a lot of the same information. While I appreciate that it's worth pointing out the difference between the Jeffreys, Bayes and Haldane priors is it really necessary to specify their posterior distributions individually? And after that there is a huge block of text which deals with the s=0 and n=0 cases in such a way that there is lots of text that says nothing. Most of this could go.
118.208.48.83 ( talk) 20:04, 24 October 2013 (UTC)
At User_talk:Jimbo_Wales#Search_recent_changes.3F, someone has pretended that page Beta distribution is owned by a terrific dragon that delete any contribution that tries to make the article clearer and shorter (see https://en.wikipedia.org/?title=User_talk%3AJimbo_Wales&type=revision&diff=710796589&oldid=710789244). Let us try if this assertion contains any part of truth. Pldx1 ( talk) 17:32, 19 March 2016 (UTC)
The inbox says x ∈ (0,1) while the first paragraph says it is [0,1]. which is it? 107.133.211.254 ( talk) 12:05, 6 August 2016 (UTC)
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The equation in the Probability Density Function section in the beginning of the article uses the parameter "u" in the denominator but fails to mention what "u" is. Clarification would be appreciated. — Preceding unsigned comment added by Pwvirgo ( talk • contribs) 15:23, 29 July 2017 (UTC)
I believe that there is an error in the following formula:
It should be the covariance between ln(X) and ln(1-X) instead of between ln(X) and 1-X:
This would be consistent with the formula in Section 3.3.1 for the 2 parameter case. However, I am not entirely sure, so perhaps someone else can also have a look at this.
2A00:E040:C0C:4501:0:0:0:167 ( talk) 10:25, 24 January 2018 (UTC)
The first sentence suggests that the distribution is a function of some random variable, which surely is not correct. I propose to remove "appear as exponents of the random variable and". — Preceding unsigned comment added by Prof. Globi ( talk • contribs) 22:31, 20 March 2019 (UTC)
The following two sections should be merged:
Tal Galili ( talk) 11:40, 30 November 2019 (UTC)
No mistake. Sorry. I don't know how to remove this comment.
The article states that in a particular limit, the Beta distribution approaches the Gamma distribution. Is there a source or derivation for this? Also, should it also mention approaching the Delta distribution in another limit? Cesiumfrog ( talk) 00:17, 4 August 2020 (UTC)
I tried to translate the article into Czech with the Wikipedia built-in translator, but I was forced to stop after a few sentences. The article is so incredibly long and equation/graphics heavy that it breaks the tool. Can please somebody knowledgeable divide the article into a set of shorter ones? There is a lot of material here which is of interest only for very specialized experts. Jan Spousta ( talk) 11:24, 22 December 2020 (UTC)
Hey :) I think the article is very rich, but that's the expected type of material about such a central distribution. It's not much different then the normal distribution, binomial distribution l, etc. Tal Galili ( talk) 18:50, 22 December 2020 (UTC)