Functions to compute the probability density function, cumulative distribution function, quantile function, and parameter estimators for the Beta distribution.
The Beta distribution is defined on \(x \in [0, 1]\) with shape parameters \(\alpha > 0\) and \(\beta > 0\).
The PDF is:
$$f(x;\alpha, \beta) = \frac{x^{\alpha - 1}(1-x)^{\beta - 1}}{B(\alpha, \beta)}$$
Where \(B(\alpha, \beta)\) is the beta function.
The CDF is:
$$F(x; \alpha, \beta) = I_x(\alpha, \beta)$$
Where \(I_x\) is the regularised incomplete beta function.
The quantile is:
$$F^{-1}(p; \alpha, \beta) = I_{p}^{-1}(\alpha, \beta)$$
Where \(I_{p}^{-1}\) is the inverse of the regularised incomplete beta function.
Usage
beta_distribution(alpha, beta)
beta_pdf(x, alpha, beta)
beta_lpdf(x, alpha, beta)
beta_cdf(x, alpha, beta)
beta_lcdf(x, alpha, beta)
beta_quantile(p, alpha, beta)
beta_find_alpha(mean = NULL, variance = NULL, beta = NULL, x = NULL, p = NULL)
beta_find_beta(mean = NULL, variance = NULL, alpha = NULL, x = NULL, p = NULL)Value
A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.
See also
Boost Documentation for more details on the mathematical background.
Examples
# Beta distribution with shape parameters alpha = 2, beta = 5
dist <- beta_distribution(2, 5)
# Apply generic functions
cdf(dist, 0.5)
#> [1] 0.890625
logcdf(dist, 0.5)
#> [1] -0.1158318
pdf(dist, 0.5)
#> [1] 0.9375
logpdf(dist, 0.5)
#> [1] -0.06453852
hazard(dist, 0.5)
#> [1] 8.571429
chf(dist, 0.5)
#> [1] 2.212973
mean(dist)
#> [1] 0.2857143
median(dist)
#> [1] 0.26445
mode(dist)
#> [1] 0.2
range(dist)
#> [1] 0 1
quantile(dist, 0.2)
#> [1] 0.1398807
standard_deviation(dist)
#> [1] 0.1597191
support(dist)
#> [1] 0 1
variance(dist)
#> [1] 0.0255102
skewness(dist)
#> [1] 0.5962848
kurtosis(dist)
#> [1] 2.88
kurtosis_excess(dist)
#> [1] -0.12
# Convenience functions
beta_pdf(0.5, 2, 5)
#> [1] 0.9375
beta_lpdf(0.5, 2, 5)
#> [1] -0.06453852
beta_cdf(0.5, 2, 5)
#> [1] 0.890625
beta_lcdf(0.5, 2, 5)
#> [1] -0.1158318
beta_quantile(0.5, 2, 5)
#> [1] 0.26445
if (FALSE) { # \dontrun{
# Find alpha given mean and variance
beta_find_alpha(mean = 0.3, variance = 0.02)
# Find alpha given beta, x, and probability
beta_find_alpha(beta = 5, x = 0.4, p = 0.6)
# Find beta given mean and variance
beta_find_beta(mean = 0.3, variance = 0.02)
# Find beta given alpha, x, and probability
beta_find_beta(alpha = 2, x = 0.4, p = 0.6)
} # }