# BetaBinom: Beta-binomial distribution In extraDistr: Additional Univariate and Multivariate Distributions

## Description

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

## Usage

 ```1 2 3 4 5``` ```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 ≤ 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 ~ Beta(α, β) and X ~ Binomial(n, p), then X ~ BetaBinomial(n, α, β).

Probability mass function

f(x) = choose(n, x) * B(x+α, n-x+β) / B(α, β)

Cumulative distribution function is calculated using recursive algorithm that employs the fact that Γ(x) = (x - 1)!, and B(x, y) = (Γ(x)Γ(y))/Γ(x+y) , and that choose(n,k) = prod((n+1-(1:k))/(1:k)) . This enables re-writing probability mass function as

f(x) = prod((n+1-(1:x))/(1:x)) * (((α+x-1)!*(β+n-x-1)!)/((α+β+n+1)!)) / B(α, β)

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

f(x+1) = prod((n+1-(1:x))/(1:x)) * ((n+1-x+1)/(x+1)) * (((α+x-1)!*(α+x)*(β+n-x-1)!/(β+n-x))/((α+β+n+1)!*(α+β+n))) / B(α, β)

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

F(x) = f(0)+...+f(x)

`Beta`, `Binomial`
 ```1 2 3 4 5 6 7 8``` ```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) ```