Editing Beta distribution (section)

===Special and limiting cases===
[[File:Random Walk example.svg|thumb|Example of eight realizations of a random walk in one dimension starting at 0: the probability for the time of the last visit to the origin is distributed as Beta(1/2, 1/2)]]
[[File:Arcsin density.svg|thumb|Beta(1/2, 1/2): The [[arcsine distribution]] probability density was proposed by [[Harold Jeffreys]] to represent uncertainty for a [[Bernoulli distribution|Bernoulli]] or a [[binomial distribution]] in [[Bayesian inference]], and is now commonly referred to as [[Jeffreys prior]]: ''p''<sup>−1/2</sup>(1&nbsp;−&nbsp;''p'')<sup>−1/2</sup>. This distribution also appears in several [[random walk]] fundamental theorems]]

* Beta(1, 1) ~ [[uniform distribution (continuous)|U(0, 1)]] with density 1 on that interval.
* Beta(n, 1) ~ Maximum of ''n'' independent rvs. with [[uniform distribution (continuous)|U(0, 1)]], sometimes called a ''a standard power function distribution'' with density ''n''&nbsp;''x''<sup>''n''–1</sup> on that interval.
* Beta(1, n) ~ Minimum of ''n'' independent rvs. with [[uniform distribution (continuous)|U(0, 1)]] with density ''n''(1&nbsp;−&nbsp;''x'')<sup>''n''−1</sup> on that interval.
* If ''X'' ~ Beta(3/2, 3/2) and ''r'' > 0 then 2''rX''&nbsp;−&nbsp;''r'' ~ [[Wigner semicircle distribution]].
* Beta(1/2, 1/2) is equivalent to the [[arcsine distribution]]. This distribution is also [[Jeffreys prior]] probability for the [[Bernoulli distribution|Bernoulli]] and  [[binomial distribution]]s.
* <math>\lim_{n \to \infty} n \operatorname{Beta}(1,n) =  \operatorname{Exponential}(1)</math>  the [[exponential distribution]].
* <math>\lim_{n \to \infty} n \operatorname{Beta}(k,n) = \operatorname{Gamma}(k,1)</math> the [[gamma distribution]].
* For large <math>n</math>, <math>\operatorname{Beta}(\alpha n,\beta n) \to \mathcal{N}\left(\frac{\alpha}{\alpha+\beta},\frac{\alpha\beta}{(\alpha+\beta)^3}\frac{1}{n}\right)</math> the [[normal distribution]]. More precisely, if <math>X_n \sim \operatorname{Beta}(\alpha n,\beta n)</math> then  <math>\sqrt{n}\left(X_n -\tfrac{\alpha}{\alpha+\beta}\right)</math> converges in distribution to a normal distribution with mean 0 and variance <math>\tfrac{\alpha\beta}{(\alpha+\beta)^3}</math> as ''n'' increases.