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 \alpha. b This is the scalar shape parameter \beta log Logical. If log=TRUE, then the logarithm of the density is returned.

### 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.

dbeta

### 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"))


[Package LaplacesDemon version 16.1.6 Index]