Talk:Beta distribution

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Contents

[edit] Identity

\int_0^1t^x(1-t)^y\,dt = \mathrm{\Beta}(x+1 \, ,y+1) = \sum_{k=0}^y {(-1)^k \, \frac {{y \choose k}}{x+1+k}}
= \frac{x! \, \, y!}{(x+y+1)!} = \frac{\Gamma(x+1) \, \, \Gamma(y+1)}{\Gamma(x+y+2)}

Ohanian 01:26, 2005 Apr 8 (UTC)

[edit] why the difference in notation?

Consider the expression

\ {n \choose i}p^i(1-p)^{n-i}

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)

This is mentioned only implicitly in the current version, which describes the beta distribution as the conjugate prior for the binomial. You could add a section on occurrence and uses of the beta distribution that would clarify this point further. --MarkSweep✍ 12:50, 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)

[edit] Should computing α and β be moved to a "Parameter Estimation" section?

Should the paragraph on computing α and β 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).

What is there now is not about parameter estimation, but it would be a short step from there to estimation by the method of moments. If there is to be a section on parameter estimation, the certainly MLEs should be there too. Also, the initial "e" in "estimation" should not be capitalized in the section heading unless it's the first letter in the section heading. Michael Hardy 22:43, 16 December 2005 (UTC)
The method of moments calculation was moved back to the intro, without any discussion. I've moved it back to the parameter estimation section. Please don't undo edits that have been discussed and agreed upon without discussing it first. User:Hilgerdenaar December 5 2006

[edit] Entropy

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)

Sorry the link was bad, I've updated it. The relevant formula is on p. 9 and is H(x) = ln[B(α,β)] − (α − 1)[ψ(α) − ψ(α + β)] − (β − 1)[ψ(β) − ψ(α + β)] where B(p,q)={\Gamma(p)\Gamma(q)\over \Gamma(p+q)}. A5 17:32, 23 May 2006 (UTC)


[edit] Organization

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.

Agreed Shae 18:27, 6 June 2007 (UTC)

[edit] Distribution Function

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.

[edit] Beta distribution of the second kind

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)

[edit] Layout

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).

The infobox on the right hand side can sometimes cause those problems. Widen your browser window, and the blank space should go away. Baccyak4H (Yak!) 20:11, 8 December 2006 (UTC)

[edit] median of beta distribution?

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

[edit] Sampling

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)

[edit] sum of independent betas

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 1F1 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)

Retraction, I found this on Wolfram as the Kummer confluent hypergeometric function of the first kind. Not too helpful, but if this is silent for awhile or if someone wants, I'll add a note and link to Confluent_hypergeometric_functions--Phays 22:25, 17 August 2007 (UTC)

[edit] Random number generator

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)