Beta prime
Probability density function
Cumulative distribution function
Parameters
α
>
0
{\displaystyle \alpha >0}
shape (real )
β
>
0
{\displaystyle \beta >0}
shape (real) Support
x
∈
[
0
,
∞
)
{\displaystyle x\in [0,\infty )\!}
PDF
f
(
x
)
=
x
α
−
1
(
1
+
x
)
−
α
−
β
B
(
α
,
β
)
{\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{B(\alpha ,\beta )}}\!}
CDF
I
x
1
+
x
(
α
,
β
)
{\displaystyle I_{{\frac {x}{1+x}}(\alpha ,\beta )}}
where
I
x
(
α
,
β
)
{\displaystyle I_{x}(\alpha ,\beta )}
is the incomplete beta function Mean
α
β
−
1
if
β
>
1
{\displaystyle {\frac {\alpha }{\beta -1}}{\text{ if }}\beta >1}
Mode
α
−
1
β
+
1
if
α
≥
1
, 0 otherwise
{\displaystyle {\frac {\alpha -1}{\beta +1}}{\text{ if }}\alpha \geq 1{\text{, 0 otherwise}}\!}
Variance
α
(
α
+
β
−
1
)
(
β
−
2
)
(
β
−
1
)
2
if
β
>
2
{\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}{\text{ if }}\beta >2}
Skewness
2
(
2
α
+
β
−
1
)
β
−
3
β
−
2
α
(
α
+
β
−
1
)
if
β
>
3
{\displaystyle {\frac {2(2\alpha +\beta -1)}{\beta -3}}{\sqrt {\frac {\beta -2}{\alpha (\alpha +\beta -1)}}}{\text{ if }}\beta >3}
MGF
e
−
t
Γ
(
α
+
β
)
Γ
(
β
)
G
1
,
2
2
,
0
(
α
+
β
β
,
0
|
−
t
)
{\displaystyle {\frac {e^{-t}\Gamma (\alpha +\beta )}{\Gamma (\beta )}}G_{1,2}^{\,2,0}\!\left(\left.{\begin{matrix}\alpha +\beta \\\beta ,0\end{matrix}}\;\right|\,-t\right)}
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
{\displaystyle x>0}
with two parameters α and β , having the probability density function :
f
(
x
)
=
x
α
−
1
(
1
+
x
)
−
α
−
β
B
(
α
,
β
)
{\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{B(\alpha ,\beta )}}}
where B is the Beta function .
The cumulative distribution function is
F
(
x
;
α
,
β
)
=
I
x
1
+
x
(
α
,
β
)
,
{\displaystyle F(x;\alpha ,\beta )=I_{\frac {x}{1+x}}\left(\alpha ,\beta \right),}
where I is the regularized incomplete beta function .
The expectation value, variance, and other details of the distribution are given in the sidebox; for
β
>
4
{\displaystyle \beta >4}
, the excess kurtosis is
γ
2
=
6
α
(
α
+
β
−
1
)
(
5
β
−
11
)
+
(
β
−
1
)
2
(
β
−
2
)
α
(
α
+
β
−
1
)
(
β
−
3
)
(
β
−
4
)
.
{\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)}}.}
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
β
′
(
α
,
β
)
{\displaystyle \beta '(\alpha ,\beta )}
is
X
^
=
α
−
1
β
+
1
{\displaystyle {\hat {X}}={\frac {\alpha -1}{\beta +1}}}
.
Its mean is
α
β
−
1
{\displaystyle {\frac {\alpha }{\beta -1}}}
if
β
>
1
{\displaystyle \beta >1}
(if
β
≤
1
{\displaystyle \beta \leq 1}
the mean is infinite, in other words it has no well defined mean) and its variance is
α
(
α
+
β
−
1
)
(
β
−
2
)
(
β
−
1
)
2
{\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}}
if
β
>
2
{\displaystyle \beta >2}
.
For
−
α
<
k
<
β
{\displaystyle -\alpha <k<\beta }
, the k -th moment
E
[
X
k
]
{\displaystyle E[X^{k}]}
is given by
E
[
X
k
]
=
B
(
α
+
k
,
β
−
k
)
B
(
α
,
β
)
.
{\displaystyle E[X^{k}]={\frac {B(\alpha +k,\beta -k)}{B(\alpha ,\beta )}}.}
For
k
∈
N
{\displaystyle k\in \mathbb {N} }
with
k
<
β
,
{\displaystyle k<\beta ,}
this simplifies to
E
[
X
k
]
=
∏
i
=
1
k
α
+
i
−
1
β
−
i
.
{\displaystyle E[X^{k}]=\prod _{i=1}^{k}{\frac {\alpha +i-1}{\beta -i}}.}
The cdf can also be written as
x
α
⋅
2
F
1
(
α
,
α
+
β
,
α
+
1
,
−
x
)
α
⋅
B
(
α
,
β
)
{\displaystyle {\frac {x^{\alpha }\cdot {}_{2}F_{1}(\alpha ,\alpha +\beta ,\alpha +1,-x)}{\alpha \cdot B(\alpha ,\beta )}}}
where
2
F
1
{\displaystyle {}_{2}F_{1}}
is the Gauss's hypergeometric function 2 F1 .
Generalization
Two more parameters can be added to form the generalized beta prime distribution .
p
>
0
{\displaystyle p>0}
shape (real )
q
>
0
{\displaystyle q>0}
scale (real )
having the probability density function :
f
(
x
;
α
,
β
,
p
,
q
)
=
p
(
x
q
)
α
p
−
1
(
1
+
(
x
q
)
p
)
−
α
−
β
q
B
(
α
,
β
)
{\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 )}}}
with mean
q
Γ
(
α
+
1
p
)
Γ
(
β
−
1
p
)
Γ
(
α
)
Γ
(
β
)
if
β
p
>
1
{\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}
and mode
q
(
α
p
−
1
β
p
+
1
)
1
p
if
α
p
≥
1
{\displaystyle q\left({\frac {\alpha p-1}{\beta p+1}}\right)^{\tfrac {1}{p}}\quad {\text{if }}\alpha p\geq 1}
Note that if p = q = 1 then the generalized beta prime distribution reduces to the standard beta prime distribution
Compound gamma distribution
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 :
β
′
(
x
;
α
,
β
,
1
,
q
)
=
∫
0
∞
G
(
x
;
α
,
r
)
G
(
r
;
β
,
q
)
d
r
{\displaystyle \beta '(x;\alpha ,\beta ,1,q)=\int _{0}^{\infty }G(x;\alpha ,r)G(r;\beta ,q)\;dr}
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 q 2 .
Properties
If
X
∼
β
′
(
α
,
β
)
{\displaystyle X\sim \beta '(\alpha ,\beta )}
then
1
X
∼
β
′
(
β
,
α
)
{\displaystyle {\tfrac {1}{X}}\sim \beta '(\beta ,\alpha )}
.
If
X
∼
β
′
(
α
,
β
,
p
,
q
)
{\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)}
then
k
X
∼
β
′
(
α
,
β
,
p
,
k
q
)
{\displaystyle kX\sim \beta '(\alpha ,\beta ,p,kq)}
.
β
′
(
α
,
β
,
1
,
1
)
=
β
′
(
α
,
β
)
{\displaystyle \beta '(\alpha ,\beta ,1,1)=\beta '(\alpha ,\beta )}
Related distributions and properties
If
X
∼
F
(
2
α
,
2
β
)
{\displaystyle X\sim F(2\alpha ,2\beta )}
has an F -distribution , then
α
β
X
∼
β
′
(
α
,
β
)
{\displaystyle {\tfrac {\alpha }{\beta }}X\sim \beta '(\alpha ,\beta )}
, or equivalently,
X
∼
β
′
(
α
,
β
,
1
,
β
α
)
{\displaystyle X\sim \beta '(\alpha ,\beta ,1,{\tfrac {\beta }{\alpha }})}
.
If
X
∼
Beta
(
α
,
β
)
{\displaystyle X\sim {\textrm {Beta}}(\alpha ,\beta )}
then
X
1
−
X
∼
β
′
(
α
,
β
)
{\displaystyle {\frac {X}{1-X}}\sim \beta '(\alpha ,\beta )}
.
If
X
∼
Γ
(
α
,
1
)
{\displaystyle X\sim \Gamma (\alpha ,1)}
and
Y
∼
Γ
(
β
,
1
)
{\displaystyle Y\sim \Gamma (\beta ,1)}
are independent, then
X
Y
∼
β
′
(
α
,
β
)
{\displaystyle {\frac {X}{Y}}\sim \beta '(\alpha ,\beta )}
.
Parametrization 1: If
X
k
∼
Γ
(
α
k
,
θ
k
)
{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\theta _{k})}
are independent, then
X
1
X
2
∼
β
′
(
α
1
,
α
2
,
1
,
θ
1
θ
2
)
{\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\theta _{1}}{\theta _{2}}})}
.
Parametrization 2: If
X
k
∼
Γ
(
α
k
,
β
k
)
{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})}
are independent, then
X
1
X
2
∼
β
′
(
α
1
,
α
2
,
1
,
β
2
β
1
)
{\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\beta _{2}}{\beta _{1}}})}
.
β
′
(
p
,
1
,
a
,
b
)
=
Dagum
(
p
,
a
,
b
)
{\displaystyle \beta '(p,1,a,b)={\textrm {Dagum}}(p,a,b)}
the Dagum distribution
β
′
(
1
,
p
,
a
,
b
)
=
SinghMaddala
(
p
,
a
,
b
)
{\displaystyle \beta '(1,p,a,b)={\textrm {SinghMaddala}}(p,a,b)}
the Singh–Maddala distribution .
β
′
(
1
,
1
,
γ
,
σ
)
=
LL
(
γ
,
σ
)
{\displaystyle \beta '(1,1,\gamma ,\sigma )={\textrm {LL}}(\gamma ,\sigma )}
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
x
m
{\displaystyle x_{m}}
and shape parameter
α
{\displaystyle \alpha }
, then
X
−
x
m
∼
β
′
(
1
,
α
)
{\displaystyle X-x_{m}\sim \beta ^{\prime }(1,\alpha )}
.
If X has a Lomax distribution , also known as a Pareto Type II distribution, with shape parameter
α
{\displaystyle \alpha }
and scale parameter
λ
{\displaystyle \lambda }
, then
X
λ
∼
β
′
(
1
,
α
)
{\displaystyle {\frac {X}{\lambda }}\sim \beta ^{\prime }(1,\alpha )}
.
If X has a standard Pareto Type IV distribution with shape parameter
α
{\displaystyle \alpha }
and inequality parameter
γ
{\displaystyle \gamma }
, then
X
1
γ
∼
β
′
(
1
,
α
)
{\displaystyle X^{\frac {1}{\gamma }}\sim \beta ^{\prime }(1,\alpha )}
, or equivalently,
X
∼
β
′
(
1
,
α
,
1
γ
,
1
)
{\displaystyle X\sim \beta ^{\prime }(1,\alpha ,{\tfrac {1}{\gamma }},1)}
.
The inverted Dirichlet distribution is a generalization of the beta prime distribution.
Notes
^ a b Johnson et al (1995), p 248
^ Dubey, Satya D. (December 1970). "Compound gamma, beta and F distributions". Metrika . 16 : 27–31. doi :10.1007/BF02613934 .
References
Discrete univariate
with finite support with infinite support
Continuous univariate
supported on a bounded interval supported on a semi-infinite interval supported on the whole real line with support whose type varies
Mixed univariate
Multivariate (joint) Directional Degenerate and singular Families