Density, distribution function, quantile function and random generation
for the beta-binomial distribution when parameterized
by the mean mu and the overdispersion parameter rho
rather than the typical shape parameters.
Usage
dbetabinom(x, size, mu, rho, log)
pbetabinom(q, size, mu, rho, log_p)
qbetabinom(p, size, mu, rho)
rbetabinom(n, size, mu, rho)Arguments
- x, q
A vector of quantiles.
- size
A vector of sizes.
- mu
Either a scalar of the mean for each observation, or a vector of means of each observation, and thus the same length as
xandsize. This must be between 0 and 1.- rho
Either a scalar of the overdispersion parameter for each observation, or a vector of overdispersion parameters of each observation, and thus the same length as
xandsize. This must be between 0 and 1.- log, log_p
A logical vector either of length 1 or the same length as
xandsize. This determines whether to return the log probabilities for all observations (in the case that its length is 1) or for each observation (in the case that its length is that ofxandsize).- p
A vector of probabilities.
- n
The number of observations.
Value
Either a random sample (rbetabinom),
the density (dbetabinom), the tail
probability (pbetabinom), or the quantile
(qbetabinom) of the beta-binomial distribution.
Details
Let \(\mu\) and \(\rho\) be the mean and overdispersion parameters. Let \(\alpha\) and \(\beta\) be the usual shape parameters of a beta distribution. Then we have the relation $$\mu = \alpha/(\alpha + \beta),$$ and $$\rho = 1/(1 + \alpha + \beta).$$ This necessarily means that $$\alpha = \mu (1 - \rho)/\rho,$$ and $$\beta = (1 - \mu) (1 - \rho)/\rho.$$