BetaBinomial | R Documentation |
Density, distribution function, quantile function, and random generation
for the beta-binomial distribution. A variable with a beta-binomial
distribution is distributed as binomial distribution with parameters
N
and p
, where the probability p
of success iteself
has a beta distribution with parameters u
and v
.
dbb(x, N, u, v, log = FALSE) pbb(q, N, u, v) qbb(p, N, u, v) rbb(n, N, u, v)
x |
vector of qauntiles |
q |
vector of quantiles |
p |
vector of probabilities |
n |
number of observations |
N |
number of trials ( a positive integer) |
u |
first positive parameter of the beta distribution |
v |
second positive parameter of the beta distribution |
log |
A logical value; if true, values are returned on the log scale |
The beta-binomial distribution with parameters N, u, and v has density given by
choose(N, x) * Beta(x + u, N - x + v) / Beta(u,v)
for u > 0, v > 0, a positive integer N, and any nonnegative integer x. Although one can express the integral in closed form using generalized hypergeometric functions, the implementation of distribution function used here simply relies on the the cumulative sum of the density.
The mean and variance of the beta-binomial distribution can be computed explicitly as
μ = \frac{Nu}{u+v}
and
σ^2 = \frac{Nuv(N+u+v)}{(u+v)^2 (1+u+v)}
dbb
gives the density, pbb
gives the distribution function,
qbb
gives the quantile function, and rbb
generates random
deviates.
Kevin R. Coombes <krc@silicovore.com>
dbeta
for the beta distribution and
dbinom
for the binomial distribution.
# set up parameters w <- 10 u <- 0.3*w v <- 0.7*w N <- 12 # generate random values from the beta-binomial x <- rbb(1000, N, u, v) # check that the empirical summary matches the theoretical one summary(x) qbb(c(0.25, 0.50, 0.75), N, u, v) # check that the empirpical histogram matches te theoretical density hist(x, breaks=seq(-0.5, N + 0.55), prob=TRUE) lines(0:N, dbb(0:N, N, u,v), type='b')
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