In probability theory and statistics , the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kind [1] absolutely continuous probability distribution defined for
x
>
0
{\displaystyle x>0}
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 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
β
′
(
α
,
β
)
{\displaystyle \beta ^{'}(\alpha ,\beta )}
X
^
=
α
−
1
β
+
1
{\displaystyle {\hat {X}}={\frac {\alpha -1}{\beta +1}}}
α
β
−
1
{\displaystyle {\frac {\alpha }{\beta -1}}}
β
>
1
{\displaystyle \beta >1}
β
≤
1
{\displaystyle \beta \leq 1}
α
(
α
+
β
−
1
)
(
β
−
2
)
(
β
−
1
)
2
{\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}}
β
>
2
{\displaystyle \beta >2}
For
−
α
<
k
<
β
{\displaystyle -\alpha <k<\beta }
E
[
X
k
]
{\displaystyle E[X^{k}]}
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} }
k
<
β
{\displaystyle k<\beta }
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}}
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}
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 (\alpha +{\tfrac {1}{p}})\Gamma (\beta -{\tfrac {1}{p}})}{\Gamma (\alpha )\Gamma (\beta )}}{\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}}{\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] 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
,
p
)
G
(
p
;
β
,
q
)
d
p
{\displaystyle \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 If
X
∼
β
′
(
α
,
β
)
{\displaystyle X\sim \beta ^{'}(\alpha ,\beta )\,}
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)\,}
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 If
X
∼
F
(
α
,
β
)
{\displaystyle X\sim F(\alpha ,\beta )\,}
α
β
X
∼
β
′
(
α
2
,
β
2
)
{\displaystyle {\tfrac {\alpha }{\beta }}X\sim \beta ^{'}({\tfrac {\alpha }{2}},{\tfrac {\beta }{2}})\,}
If
X
∼
Beta
(
α
,
β
)
{\displaystyle X\sim {\textrm {Beta}}(\alpha ,\beta )\,}
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)\,}
Y
∼
Γ
(
β
,
1
)
{\displaystyle Y\sim \Gamma (\beta ,1)\,}
X
Y
∼
β
′
(
α
,
β
)
{\displaystyle {\frac {X}{Y}}\sim \beta ^{'}(\alpha ,\beta )}
β
′
(
p
,
1
,
a
,
b
)
=
Dagum
(
p
,
a
,
b
)
{\displaystyle \beta ^{'}(p,1,a,b)={\textrm {Dagum}}(p,a,b)\,}
Dagum distribution
β
′
(
1
,
p
,
a
,
b
)
=
SinghMaddala
(
p
,
a
,
b
)
{\displaystyle \beta ^{'}(1,p,a,b)={\textrm {SinghMaddala}}(p,a,b)\,}
Singh Maddala distribution
β
′
(
1
,
1
,
γ
,
σ
)
=
LL
(
γ
,
σ
)
{\displaystyle \beta ^{'}(1,1,\gamma ,\sigma )={\textrm {LL}}(\gamma ,\sigma )\,}
Log logistic distribution Beta prime distribution is a special case of the type 6 Pearson distribution
Pareto distribution type II is related to Beta prime distributionPareto distribution type IV is related to Beta prime distributionNotes References Jonhnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariate Distributions , Volume 2 (2nd Edition), Wiley. ISBN 0-471-58494-0
Discrete
with finite with infinite
Continuous
supported on a supported on a supported with support
Mixed
Multivariate Directional Degenerate singular Families
![{\displaystyle E[X^{k}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dcd54fe6c5cb4afbfcd7bd94c4778d13b8bbc3f)
![{\displaystyle E[X^{k}]={\frac {B(\alpha +k,\beta -k)}{B(\alpha ,\beta )}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30c6530cfd83409026129cc40968169281f41081)
![{\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)
