Probability distribution
Beta prime Probability density function ![](//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Beta_prime_pdf.svg/325px-Beta_prime_pdf.svg.png) |
Cumulative distribution function ![](//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Beta_prime_cdf.svg/325px-Beta_prime_cdf.svg.png) |
Parameters | shape (real)
shape (real) |
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Support | ![{\displaystyle x\in [0,\infty )\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be00d03e32dc2701e685f3f746dee4702dfdd208) |
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PDF | ![{\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{B(\alpha ,\beta )}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b14e2ca05e0237f7b98dca313f3c3f8658fe1a4) |
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CDF | where is the incomplete beta function |
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Mean | ![{\displaystyle {\frac {\alpha }{\beta -1}}{\text{ if }}\beta >1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e26f4eb2c328cfddbaae3bac89e74e596b29c2a) |
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Mode | ![{\displaystyle {\frac {\alpha -1}{\beta +1}}{\text{ if }}\alpha \geq 1{\text{, 0 otherwise}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6317382f881af2a06b1acd41dbccc9f6effcc97d) |
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Variance | ![{\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}{\text{ if }}\beta >2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b295ffdae2069977c7e9b4d1347e331314e22d0) |
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Skewness | ![{\displaystyle {\frac {2(2\alpha +\beta -1)}{\beta -3}}{\sqrt {\frac {\beta -2}{\alpha (\alpha +\beta -1)}}}{\text{ if }}\beta >3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e81a86b913ff240a54f87ecf4c01bb633e994bc) |
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MGF | Does not exist |
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CF | ![{\displaystyle {\frac {e^{-it}\Gamma (\alpha +\beta )}{\Gamma (\beta )}}G_{1,2}^{\,2,0}\!\left(\left.{\begin{matrix}\alpha +\beta \\\beta ,0\end{matrix}}\;\right|\,-it\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d02d27cf5d3f60e5dafdfa9384560412c29bc955) |
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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. If
has a beta distribution, then the odds
has a beta prime distribution.
Definitions
Beta prime distribution is defined for
with two parameters α and β, having the probability density function:
![{\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{B(\alpha ,\beta )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef5a50a437e1979e33821c36de0701d63164002)
where B is the Beta function.
The cumulative distribution function is
![{\displaystyle F(x;\alpha ,\beta )=I_{\frac {x}{1+x}}\left(\alpha ,\beta \right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f38285fb9c37e5ab4e4264fe836ee9a752735ca2)
where I is the regularized incomplete beta function.
The expected value, variance, and other details of the distribution are given in the sidebox; for
, the excess kurtosis is
![{\displaystyle \gamma _{2}=6{\frac {\alpha (\alpha +\beta -1)(5\beta -11)+(\beta -1)^{2}(\beta -2)}{\alpha (\alpha +\beta -1)(\beta -3)(\beta -4)}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5fe477361711a90e85f58db17d7b1e9f6da8b1c)
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
is
. Its mean is
if
(if
the mean is infinite, in other words it has no well defined mean) and its variance is
if
.
For
, the k-th moment
is given by
![{\displaystyle E[X^{k}]={\frac {B(\alpha +k,\beta -k)}{B(\alpha ,\beta )}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30c6530cfd83409026129cc40968169281f41081)
For
with
this simplifies to
![{\displaystyle E[X^{k}]=\prod _{i=1}^{k}{\frac {\alpha +i-1}{\beta -i}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0f1689a0ef95460a83f9f53462da32a9b1e8f04)
The cdf can also be written as
![{\displaystyle {\frac {x^{\alpha }\cdot {}_{2}F_{1}(\alpha ,\alpha +\beta ,\alpha +1,-x)}{\alpha \cdot B(\alpha ,\beta )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee46eb9610755a18e61957792780d5d9fb25f109)
where
is the Gauss's hypergeometric function 2F1 .
Alternative parameterization
The beta prime distribution may also be reparameterized in terms of its mean μ > 0 and precision ν > 0 parameters ([2] p. 36).
Consider the parameterization μ = α/(β-1) and ν = β- 2, i.e., α = μ( 1 + ν) and β = 2 + ν. Under this parameterization E[Y] = μ and Var[Y] = μ(1 + μ)/ν.
Generalization
Two more parameters can be added to form the generalized beta prime distribution
:
shape (real)
scale (real)
having the probability density function:
![{\displaystyle 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 )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b5f1f8aa8bd75f397eecda90000f31fdeb7fd4e)
with mean
![{\displaystyle {\frac {q\Gamma \left(\alpha +{\tfrac {1}{p}}\right)\Gamma (\beta -{\tfrac {1}{p}})}{\Gamma (\alpha )\Gamma (\beta )}}\quad {\text{if }}\beta p>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e22ef0e756cac49fc4b9774ad62337cecb88a4b)
and mode
![{\displaystyle q\left({\frac {\alpha p-1}{\beta p+1}}\right)^{\tfrac {1}{p}}\quad {\text{if }}\alpha p\geq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6c8881c6a2ff5bc28f423af9d2996610d196468)
Note that if p = q = 1 then the generalized beta prime distribution reduces to the standard beta prime distribution.
This generalization can be obtained via the following invertible transformation. If
and
for
, then
.
Compound gamma distribution
The compound gamma distribution[3] 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:
![{\displaystyle \beta '(x;\alpha ,\beta ,1,q)=\int _{0}^{\infty }G(x;\alpha ,r)G(r;\beta ,q)\;dr}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cad54ba3607d1cf04cc5ff382db4f3e9b6c4f91)
where
is the gamma pdf with shape
and inverse scale
.
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.
Another way to express the compounding is if
and
, then
. (This gives one way to generate random variates with compound gamma, or beta prime distributions. Another is via the ratio of independent gamma variates, as shown below.)
Properties
- If
then
. - If
, and
, then
. - If
then
. ![{\displaystyle \beta '(\alpha ,\beta ,1,1)=\beta '(\alpha ,\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a97057eff9376f94c89707d1e3bce0e0067b318)
- If
and
two iid variables, then
with
and
, as the beta prime distribution is infinitely divisible. - More generally, let
iid variables following the same beta prime distribution, i.e.
, then the sum
with
and
.
- If
has an F-distribution, then
, or equivalently,
. - If
then
. - If
then
. - For gamma distribution parametrization I:
- If
are independent, then
. Note
are all scale parameters for their respective distributions.
- For gamma distribution parametrization II:
- If
are independent, then
. The
are rate parameters, while
is a scale parameter. - If
and
, then
. The
are rate parameters for the gamma distributions, but
is the scale parameter for the beta prime.
the Dagum distribution
the Singh–Maddala distribution.
the log logistic distribution. - The beta prime distribution is a special case of the type 6 Pearson distribution.
- If X has a Pareto distribution with minimum
and shape parameter
, then
. - If X has a Lomax distribution, also known as a Pareto Type II distribution, with shape parameter
and scale parameter
, then
. - If X has a standard Pareto Type IV distribution with shape parameter
and inequality parameter
, then
, or equivalently,
. - The inverted Dirichlet distribution is a generalization of the beta prime distribution.
- If
, then
has a generalized logistic distribution. More generally, if
, then
has a scaled and shifted generalized logistic distribution.
Notes
- ^ a b Johnson et al (1995), p 248
- ^ Bourguignon, M.; Santos-Neto, M.; de Castro, M. (2021). "A new regression model for positive random variables with skewed and long tail". Metron. 79: 33–55. doi:10.1007/s40300-021-00203-y. S2CID 233534544.
- ^ Dubey, Satya D. (December 1970). "Compound gamma, beta and F distributions". Metrika. 16: 27–31. doi:10.1007/BF02613934. S2CID 123366328.
References
- Johnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd Edition), Wiley. ISBN 0-471-58494-0
- Bourguignon, M.; Santos-Neto, M.; de Castro, M. (2021), "A new regression model for positive random variables with skewed and long tail", Metron, 79: 33–55, doi:10.1007/s40300-021-00203-y, S2CID 233534544
Discrete univariate | with finite support | |
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with infinite support | |
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Continuous univariate | supported on a bounded interval | |
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supported on a semi-infinite interval | |
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supported on the whole real line | |
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with support whose type varies | |
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Mixed univariate | |
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Multivariate (joint) | |
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Directional | |
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Degenerate and singular | |
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Families | |
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Category Commons |