# The Mean-Precision Parameterized Beta Distribution Density function, distribution function, quantile function, and random generation for the (augmented) beta distribution with the mean-precision parameterization. ```r dBeta(x, mu, phi, q0 = NULL, q1 = NULL, log = FALSE) qBeta(prob, mu, phi, q0 = NULL, q1 = NULL, log.prob = FALSE) pBeta(q, mu, phi, q0 = NULL, q1 = NULL, log.prob = FALSE) rBeta(n, mu, phi, q0 = NULL, q1 = NULL) ``` ## Arguments - `x, q`: a vector of quantiles. - `mu`: the mean parameter. It must lie in (0, 1). - `phi`: the precision parameter. It must be a real positive value. - `q0`: the probability of augmentation in zero. It must lie in (0, 1). In case of no augmentation, it is `NULL` (default). - `q1`: the probability of augmentation in one. It must lie in (0, 1). In case of no augmentation, it is `NULL` (default). - `log`: logical; if TRUE, densities are returned on log-scale. - `prob`: a vector of probabilities. - `log.prob`: logical; if TRUE, probabilities `prob` are given as log(prob). - `n`: the number of values to generate. If `length(n) > 1`, the length is taken to be the number required. ## Returns The function `dBeta` returns a vector with the same length as `x` containing the density values. The function `pBeta` returns a vector with the same length as `q` containing the values of the distribution function. The function `qBeta` returns a vector with the same length as `prob` containing the quantiles. The function `rBeta` returns a vector of length `n` containing the generated random values. ## Details The beta distribution has density $$ f_B(x;\mu,\phi)=\frac{\Gamma{(\phi)}}{\Gamma{(\mu\phi)}\Gamma{((1-\mu)\phi)}}x^{\mu\phi-1}(1-x)^{(1-\mu)\phi-1} $$ for $00$ is the precision parameter. The augmented beta distribution has density * $q_0$, if $x=0$ * $q_1$, if $x=1$ * $(1-q_0-q_1)f_B(x;\mu,\phi)$, if $0\

Beta function