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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^[1].

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.

Notes

^ ^a ^b Johnson et al (1995), p248

References

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

MathWorld article

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Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

Boltzmann · Conway-Maxwell-Poisson · compound Poisson · discrete phase-type · extended negative binomial · Gauss-Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule-Simon · zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Beta · Irwin-Hall · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Beta prime · Burr · chi-square · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scale inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (-∞,∞)

Cauchy · extreme value · exponential power · Fisher's z · Fisher-Tippett · generalized hyperbolic · hyperbolic secant · Landau · Laplace · Lévy skew alpha-stable · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · Pearson · Tweedie

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