dist.Inverse.Beta {LaplacesDemon} | R Documentation |

## Inverse Beta Distribution

### Description

This is the density function and random generation from the inverse beta distribution.

### Usage

```
dinvbeta(x, a, b, log=FALSE)
rinvbeta(n, a, b)
```

### Arguments

`n` |
This is the number of draws from the distribution. |

`x` |
This is a location vector at which to evaluate density. |

`a` |
This is the scalar shape parameter |

`b` |
This is the scalar shape parameter |

`log` |
Logical. If |

### Details

Application: Continuous Univariate

Density:

`p(\theta) = \frac{\theta^{\alpha - 1} (1 + \theta)^{-\alpha - \beta}}{\beta(\alpha, \beta)}`

Inventor: Dubey (1970)

Notation 1:

`\theta \sim \mathcal{B}^{-1}(\alpha, \beta)`

Notation 2:

`p(\theta) = \mathcal{B}^{-1}(\theta | \alpha, \beta)`

Parameter 1: shape

`\alpha > 0`

Parameter 2: shape

`\beta > 0`

Mean:

`E(\theta) = \frac{\alpha}{\beta - 1}`

, for`\beta > 1`

Variance:

`var(\theta) = \frac{\alpha(\alpha + \beta - 1)}{(\beta - 1)^2 (\beta - 2)}`

Mode:

`mode(\theta) = \frac{\alpha - 1}{\beta + 1}`

The inverse-beta, also called the beta prime distribution, applies to variables that are continuous and positive. The inverse beta is the conjugate prior distribution of a parameter of a Bernoulli distribution expressed in odds.

The inverse-beta distribution has also been extended to the generalized beta prime distribution, though it is not (yet) included here.

### Value

`dinvbeta`

gives the density and
`rinvbeta`

generates random deviates.

### References

Dubey, S.D. (1970). "Compound Gamma, Beta and F Distributions".
*Metrika*, 16, p. 27–31.

### See Also

### Examples

```
library(LaplacesDemon)
x <- dinvbeta(5:10, 2, 3)
x <- rinvbeta(10, 2, 3)
#Plot Probability Functions
x <- seq(from=0.1, to=20, by=0.1)
plot(x, dinvbeta(x,2,2), ylim=c(0,1), type="l", main="Probability Function",
ylab="density", col="red")
lines(x, dinvbeta(x,2,3), type="l", col="green")
lines(x, dinvbeta(x,3,2), type="l", col="blue")
legend(2, 0.9, expression(paste(alpha==2, ", ", beta==2),
paste(alpha==2, ", ", beta==3), paste(alpha==3, ", ", beta==2)),
lty=c(1,1,1), col=c("red","green","blue"))
```

*LaplacesDemon*version 16.1.6 Index]