Beta prime distribution

From Wikipedia, the free encyclopedia

Beta Prime
Probability density function
Cumulative distribution function
Parameters α > 0 shape (real)
β > 0 shape (real)
Support x > 0\!
Probability density function (pdf) f(x) = \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}\!
Cumulative distribution function (cdf) \frac{x^\alpha \cdot _2F_1(\alpha, \alpha+\beta, \alpha+1, -x)}{\alpha \cdot B(\alpha,\beta)}\!

where 2F1 is the Gauss's hypergeometric function 2F1
Mean \frac{\alpha}{\beta-1}
Median
Mode \frac{\alpha-1}{\beta+1}\!
Variance \frac{\alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^2}
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

A Beta Prime Distribution is a probability distribution defined for x>0 with two parameters (of positive real part), α and β, having the probability density function:

f(x) = \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}

where B is a Beta function. This distribution is also known[1] as the beta distribution of the second kind. It is basically the same as the F distribution--if b is distributed as the beta prime distribution Beta'(α,β), then bβ/α obeys the F distribution with 2α and 2β degrees of freedom. The distribution is a Pearson type VI distribution[2].

The mode of a variate X distributed as β'(α,β) is \hat{X} = \frac{\alpha-1}{\beta+1}. Its mean is \frac{\alpha}{\beta-1} if β > 1 (if β < = 1 the mean is infinite, in other words it has no well defined mean) and its variance is \frac{\alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^2} if β > 2.

If X is a β'(α,β) variate then \frac{1}{X} is a β'(β,α) variate.

If X is a β(α,β) then \frac{1-X}{X} and \frac{X}{1-X} are β'(β,α) and β'(α,β) variates.

If X and Y are γ(α1) and γ(α2) variates, then \frac{X}{Y} is a β'(α1,α2) variate.

[edit] Notes

  1. ^ Johnson et al (1995), p248
  2. ^ Johnson et al (1995), p248

[edit] References

Jonhnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariuate Distributions, Volume 2 (2nd Edition), Wiley. ISBN 0-471-58494-0

MathWorld article

v • d • e
Probability distributions
 
Continuous univariate supported on a semi-infinite interval
Languages