Inverse beta distribution

The inverse beta distribution is a univariate distribution for continuous random variables , with two parameters and . It is a special case of the gamma-gamma distribution and thus a mixed distribution . ${\displaystyle \alpha }$ ${\displaystyle \beta }$

The density function is:

${\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{\mathrm {B} (\alpha ,\beta )}}}$.

This is the beta function . ${\displaystyle \mathrm {B} (\alpha ,\beta )}$

A random variable that follows an inverse beta distribution has the expected value ${\displaystyle X}$

${\displaystyle \operatorname {E} (X)={\frac {\alpha }{\beta -1}}{\text{, falls }}\beta >1}$

the mode

${\displaystyle \operatorname {Mod} (X)={\frac {\alpha -1}{\beta +1}}{\text{, falls }}\alpha \geq 1{\text{, sonst }}\operatorname {Mod} (X)=0}$

and the variance

${\displaystyle \operatorname {Var} (X)={\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}{\text{, falls }}\beta >2}$.

Relationship to the gamma distribution

If the second parameter of the gamma distribution is a random variable that is distributed like a gamma distribution, then the resulting random variable follows an inverse beta distribution . ${\displaystyle \epsilon }$ ${\displaystyle {\mathcal {G}}(a,\epsilon )}$ ${\displaystyle {\mathcal {G}}(b,1)}$ ${\displaystyle {\mathcal {InvB}}(a,b)}$

Relationship to the gamma-gamma distribution

A gamma-gamma distribution corresponds to an inverse beta distribution . ${\displaystyle \operatorname {Gamma-Gamma} (a,b=1,d)}$ ${\displaystyle {\mathcal {InvB}}(\alpha =d,\beta =a)}$

Web links

• Eric W. Weisstein : Beta Prime Distribution . In: MathWorld (English).