Inverse Beta Distribution
This is the density function and random generation from the inverse beta distribution.
dinvbeta(x, a, b, log=FALSE) rinvbeta(n, a, b)
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 |
Application: Continuous Univariate
Density: (theta^(alpha - 1) * (1 + theta)^(-alpha - beta)) / beta(alpha, beta)
Inventor: Dubey (1970)
Notation 1: theta ~ B^-1(alpha, beta)
Notation 2: p(theta) = B^-1(theta | alpha, beta)
Parameter 1: shape alpha > 0
Parameter 2: shape beta > 0
Mean: E(theta) = alpha / (beta - 1), for beta > 1
Variance: var(theta) = (alpha * (alpha + beta - 1)) / ((beta - 1)^2 * (beta - 2))
Mode: mode(theta) = (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.
dinvbeta gives the density and
rinvbeta generates random deviates.
Dubey, S.D. (1970). "Compound Gamma, Beta and F Distributions". Metrika, 16, p. 27–31.
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"))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.