extraDistr (version 1.10.0)

BetaBinom: Beta-binomial distribution

Description

Probability mass function and random generation for the beta-binomial distribution.

Usage

dbbinom(x, size, alpha = 1, beta = 1, log = FALSE)
pbbinom(q, size, alpha = 1, beta = 1, lower.tail = TRUE, log.p = FALSE)
rbbinom(n, size, alpha = 1, beta = 1)

Arguments

x, q

vector of quantiles.

size

number of trials (zero or more).

alpha, beta

non-negative parameters of the beta distribution.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are $P[X\le x]$ otherwise, $P[X>x]$.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If $p\sim \mathrm{Beta}(\alpha ,\beta )$ and $X\sim \mathrm{Binomial}(n,p)$, then $X\sim \mathrm{BetaBinomial}(n,\alpha ,\beta )$.

Probability mass function $$f(x)=(\genfrac{}{}{0ex}{}{n}{x})\frac{\mathrm{B}(x+\alpha ,n-x+\beta )}{\mathrm{B}(\alpha ,\beta )}$$

Cumulative distribution function is calculated using recursive algorithm that employs the fact that $\mathrm{\Gamma }(x)=(x-1)!$, and $\mathrm{B}(x,y)=\frac{\mathrm{\Gamma }(x)\mathrm{\Gamma }(y)}{\mathrm{\Gamma }(x+y)}$, and that $(\genfrac{}{}{0ex}{}{n}{k})=\prod _{i=1}^k\frac{n+1-i}{i}$. This enables re-writing probability mass function as

$$f(x)=\left(\prod _{i=1}^x\frac{n+1-i}{i}\right)\frac{\frac{(\alpha +x-1)!(\beta +n-x-1)!}{(\alpha +\beta +n-1)!}}{\mathrm{B}(\alpha ,\beta )}$$

what makes recursive updating from $x$ to $x+1$ easy using the properties of factorials

$$f(x+1)=\left(\prod _{i=1}^x\frac{n+1-i}{i}\right)\frac{n+1-x+1}{x+1}\frac{\frac{(\alpha +x-1)!(\alpha +x)(\beta +n-x-1)!(\beta +n-x{)}^{-1}}{(\alpha +\beta +n-1)!(\alpha +\beta +n)}}{\mathrm{B}(\alpha ,\beta )}$$

and let's us efficiently calculate cumulative distribution function as a sum of probability mass functions

$$F(x)=\sum _{k=0}^{x}f(k)$$

See Also

Beta, Binomial

Examples

x <- rbbinom(1e5, 1000, 5, 13)
xx <- 0:1000
hist(x, 100, freq = FALSE)
lines(xx-0.5, dbbinom(xx, 1000, 5, 13), col = "red")
hist(pbbinom(x, 1000, 5, 13))
xx <- seq(0, 1000, by = 0.1)
plot(ecdf(x))
lines(xx, pbbinom(xx, 1000, 5, 13), col = "red", lwd = 2)