Beta Probability Distribution Family

A Beta Probability Distribution Family is a probability density function family [math]\displaystyle{ Beta(x;α,β) }[/math] two shape parameters α and β that ...

• Context:
  • It can be instantiated as a Beta Density Function.
  • …
• Example(s):
  • http://en.wikipedia.org/wiki/File:Beta_distribution_pdf.png
  • …
• Counter-Example(s):
  • Beta-Binomial Distribution, or a Beta Prime Distribution.
  • a Gamma Distribution Familiy.
• See: Confluent Hypergeometric Function, Digamma Function, Trigamma Function, Fisher Information Matrix, Binomial Process.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/beta_distribution Retrieved:2015-6-21.
  • In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parametrized by two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution. The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. For example, it has been used as a statistical description of allele frequencies in population genetics; time allocation in project management / control systems; sunshine data; variability of soil properties; proportions of the minerals in rocks in stratigraphy; and heterogeneity in the probability of HIV transmission. In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions. For example, the beta distribution can be used in Bayesian analysis to describe initial knowledge concerning probability of success such as the probability that a space vehicle will successfully complete a specified mission. The beta distribution is a suitable model for the random behavior of percentages and proportions.

    The usual formulation of the beta distribution is also known as the beta distribution of the first kind, whereas beta distribution of the second kind is an alternative name for the beta prime distribution.

http://en.wikipedia.org/wiki/File:Beta_distribution_pdf.png

2008

• (Upton & Cook, 2008) ⇒ Graham Upton, and Ian Cook. (2008). “A Dictionary of Statistics, 2nd edition revised." Oxford University Press. ISBN:0199541450
  • QUOTE: A distribution often used as a prior distribution for a proportion. The probability density function, for a random variable X having a beta distribution is :[math]\displaystyle{ f(x) = {\frac{1}{B(\alpha \beta)} }{x^{\alpha - 1} } (1 - x)^{\beta - 1} }[/math] :[math]\displaystyle{ 0 \lt x \lt 1 }[/math], where [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math] are positive parameters, and B is the beta function. The name appears in a 1911 publication by Gini. The distribution has mean :[math]\displaystyle{ \frac {\alpha} {\alpha + \beta'} }[/math], If both [math]\displaystyle{ \alpha \lt 1 }[/math] and [math]\displaystyle{ \beta \gt 1 }[/math], the distribution has mode at :[math]\displaystyle{ \alpha \beta \over {(\alpha + \beta)^{2}(\alpha + \beta + 1)} }[/math] If both [math]\displaystyle{ \alpha \lt 1 }[/math] and [math]\displaystyle{ \beta \gt 1 }[/math], then the distribution is U-shaped, whereas, if just one of a and B is < 1, then the distributionis I-shaped. If [math]\displaystyle{ Y_l, Y_2, ..., Y_k }[/math] are independent random variables, with [math]\displaystyle{ Y_j }[/math]. having a chi-squared distribution with [math]\displaystyle{ v_j }[/math] degrees of freedom, then the ratio :[math]\displaystyle{ \sum_{j=1}^{k - 1}Y_j \bigg / \sum_{j=1}^{k}Y_j }[/math] has a distribution with [math]\displaystyle{ \alpha = \frac{1}{2} \sum_{j=1}^{k-1}v_j }[/math] and [math]\displaystyle{ \beta = \frac {1}{2}v_k }[/math]

2006

• (Dubnicka, 2006g) ⇒ Suzanne R. Dubnicka. (2006). “Special Continuous Distributions - Handout 7." Kansas State University, Introduction to Probability and Statistics I, STAT 510 - Fall 2006.
  • TERMINOLOGY : A random variable X is said to have a beta distribution with parameters a > 0 and b > 0 if its pdf is given by fX(x) =
    • (􀀀(a+b) 􀀀(a)􀀀(b)xa−1(1 − x)b−1, 0 < x < 1
    • 0, otherwise.

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