Beta prime distribution

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Beta Prime
Probability density function
Beta prime pdf.svg
Cumulative distribution function
Beta prime cdf.svg
Parameters \alpha > 0 shape (real)
\beta > 0 shape (real)
Support x > 0\!
pdf f(x) = \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}\!
CDF  I_{\frac{x}{1+x}(\alpha,\beta) } where I_x(\alpha,\beta) is the incomplete beta function
Mean \frac{\alpha}{\beta-1} \text{ if } \beta>1
Mode \frac{\alpha-1}{\beta+1} \text{ if } \alpha\ge 1\text{, 0 otherwise}\!
Variance \frac{\alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^2} \text{ if } \beta>2
Skewness \frac{2(2\alpha+\beta-1)}{\beta-3}\sqrt{\frac{\beta-2}{\alpha(\alpha+\beta-1)}} \text{ if } \beta>3

In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kind[1]) is an absolutely continuous probability distribution defined for x > 0 with two parameters 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. While the related beta distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed as a probability, the beta prime distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed in odds. The distribution is a Pearson type VI distribution.[1]

The mode of a variate X distributed as \beta^{'}(\alpha,\beta) is \hat{X} = \frac{\alpha-1}{\beta+1}. Its mean is \frac{\alpha}{\beta-1} if \beta>1 (if \beta \leq 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 \beta>2.

For -\alpha <k <\beta , the k-th moment  E[X^k] is given by

 E[X^k]=\frac{B(\alpha+k,\beta-k)}{B(\alpha,\beta)}.

For  k\in \mathbb{N} with k <\beta , this simplifies to

 E[X^k]=\prod_{i=1}^{k}  \frac{\alpha+i-1}{\beta-i}.

The cdf can also be written as

 \frac{x^\alpha \cdot _2F_1(\alpha, \alpha+\beta, \alpha+1, -x)}{\alpha \cdot B(\alpha,\beta)}\!

where _2F_1 is the Gauss's hypergeometric function 2F1 .


Differential equation


\left\{\left(x^2+x\right) f'(x)+f(x) (-p+q
   x+x+1)=0,f(1)=\frac{2^{-p-q}}{B(p,q)}\right\}


Generalization[edit]

Two more parameters can be added to form the generalized beta prime distribution.

p > 0 shape (real)
q > 0 scale (real)

having the probability density function:

f(x;\alpha,\beta,p,q) = \frac{p{\left({\frac{x}{q}}\right)}^{\alpha p-1} \left({1+{\left({\frac{x}{q}}\right)}^p}\right)^{-\alpha -\beta}}{qB(\alpha,\beta)}

with mean

\frac{q\Gamma(\alpha+\tfrac{1}{p})\Gamma(\beta-\tfrac{1}{p})}{\Gamma(\alpha)\Gamma(\beta)} \text{ if } \beta p>1

and mode

q{\left({\frac{\alpha p -1}{\beta p +1}}\right)}^\tfrac{1}{p} \text{ if } \alpha p\ge 1\!

Note that if p=q=1 then the generalized beta prime distribution reduces to the standard beta prime distribution

Compound gamma distribution[edit]

The compound gamma distribution[2] is the generalization of the beta prime when the scale parameter, q is added, but where p=1. It is so named because it is formed by compounding two gamma distributions:

\beta'(x;\alpha,\beta,1,q) = \int_0^\infty G(x;\alpha,p)G(p;\beta,q) \; dp

where G(x;a,b) is the gamma distribution with shape a and inverse scale b. This relationship can be used to generate random variables with a compound gamma, or beta prime distribution.

The mode, mean and variance of the compound gamma can be obtained by multiplying the mode and mean in the above infobox by q and the variance by q2.

Properties[edit]

  • If X \sim \beta^{'}(\alpha,\beta)\, then \tfrac{1}{X} \sim \beta^{'}(\beta,\alpha).
  • If X \sim \beta^{'}(\alpha,\beta,p,q)\, then kX \sim \beta^{'}(\alpha,\beta,p,kq)\,.
  • \beta^{'}(\alpha,\beta,1,1) = \beta^{'}(\alpha,\beta)\,

Related distributions[edit]

Notes[edit]

  1. ^ a b Johnson et al (1995), p248
  2. ^ Dubey, Satya D. (December 1970). "Compound gamma, beta and F distributions". Metrika 16: 27–31. doi:10.1007/BF02613934. 

References[edit]