# Beta prime distribution

Probability density function | |

Cumulative distribution function | |

Parameters |
shape (real) shape (real) |
---|---|

Support | |

CDF | where is the incomplete beta function |

Mean | |

Mode | |

Variance | |

Skewness |

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

where *B* is a Beta function.

The cumulative distribution function is

where *I* is the regularized incomplete beta function.

The expectation value, variance, and other details of the distribution are given in the sidebox; for , the excess kurtosis is

- .

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

For with , this simplifies to

The cdf can also be written as

where is the Gauss's hypergeometric function _{2}F_{1} .

## Generalization

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

having the probability density function:

with mean

and mode

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:

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 then .
- If then .

## Related distributions and properties

- If then , or equivalently,
- If then
- If and are independent, then .
- Parametrization 1: If are independent, then
- Parametrization 2: If are independent, then
- the Dagum distribution
- the Singh-Maddala distribution
- the 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 distribution
- Pareto distribution type IV is related to Beta prime distribution
- inverted Dirichlet distribution, a generalization of the beta prime distribution

## Notes

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

## References

- Jonhnson, N.L., Kotz, S., Balakrishnan, N. (1995).
*Continuous Univariate Distributions*, Volume 2 (2nd Edition), Wiley. ISBN 0-471-58494-0 - MathWorld article