beta prime distribution について

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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〔Johnson et al (1995), p248〕) is an absolutely continuous probability distribution defined for

x > 0

with two parameters α and β, having the probability density function:
:

f(x) = \frac}

where ''B'' is a Beta function.
The cumulative distribution function is
:

F(x; \alpha,\beta)=I_}\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

\beta>4

, the excess kurtosis is
:

\gamma_2 = 6\frac

.
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.〔
The mode of a variate ''X'' distributed as

\beta^(\alpha,\beta)

\hat = \frac

.
Its mean is

\frac

\beta>1

(if

\beta \leq 1

the mean is infinite, in other words it has no well defined mean)
and its variance is

\frac

\beta>2

.
For

-\alpha, the k-th moment E() is given by

)=\frac.

For

k\in \mathbb

with

k <\beta

, this simplifies to
:

)=\prod_^ \frac.

The cdf can also be written as
:

\frac\!

where

_2F_1

is the Gauss's hypergeometric function ₂F₁ .
Differential equation

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==Generalization==

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}\right)}^ \left(}\right)}^p}\right)^}

with mean
:

\frac)\Gamma(\beta-\tfrac)} \text \beta p>1

and mode
:

q}\right)}^\tfrac \text \alpha p\ge 1\!

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

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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