Beta prime distribution

Probability distribution
name =Beta Prime| type =density
pdf_

cdf_

parameters =$alpha\; >\; 0$ shape (real)
shape (real)
support =$x\; >\; 0!$
pdf ={B(alpha,eta)}!
cdf =where $\_2F\_1$ is the Gauss's hypergeometric function 2F1
mean =
median =
mode =
variance =
skewness =
kurtosis =
entropy =
mgf =
char =
A Beta Prime Distribution is a probability distribution defined for x>0 with two parameters (of positive real part), α and β, having the probability density function:

where $B$ is a Beta function. This distribution is also known [Johnson et al (1995), p248] as the beta distribution of the second kind. It is basically the same as the F distribution--if b is distributed as the beta prime distribution Beta'(α,β), then bβ/α obeys the F distribution with 2α and 2β degrees of freedom. The distribution is a Pearson type VI distribution [Johnson et al (1995), p248] .

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 .

If X is a variate then $frac\{1\}\{X\}$ is a variate.

If X is a then $frac\{1-X\}\{X\}$ and $frac\{X\}\{1-X\}$ are and variates.

If X and Y are $gamma(alpha\_1)$ and $gamma(alpha\_2)$ variates, then $frac\{X\}\{Y\}$ is a variate.

Notes

References

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

[http://mathworld.wolfram.com/BetaPrimeDistribution.html MathWorld article]