BetaBinom

0th

Percentile

Beta-binomial distribution

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

Keywords
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 $$ f(x) = {n \choose x} \frac{\mathrm{B}(x+\alpha, n-x+\beta)}{\mathrm{B}(\alpha, \beta)} $$

Cumulative distribution function is calculated using recursive algorithm that employs the fact that \(\Gamma(x) = (x - 1)!\), and \( \mathrm{B}(x, y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \), and that \( {n \choose k} = \prod_{i=1}^k \frac{n+1-i}{i} \). 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

Aliases
  • BetaBinom
  • dbbinom
  • pbbinom
  • rbbinom
Examples
# NOT RUN {
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)

# }
Documentation reproduced from package extraDistr, version 1.8.11, License: GPL-2

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