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Noncentral Beta Distribution Functions

Source: R/non_central_beta_distribution.R
non_central_beta_distribution.Rd

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Noncentral Beta distribution.

The noncentral beta distribution is a generalization of the Beta Distribution.

The PDF is: $$f(x; \alpha, \beta, \lambda) = \sum_{i=0}^{\infty} P(i; \lambda/2) I'_x(\alpha+i, \beta)$$ where \(P(i; \lambda/2)\) is the discrete Poisson probability at \(i\), with mean \(\lambda/2\), and \(I'_x(\alpha, \beta)\) is the derivative of the incomplete beta function.

The CDF is: $$F(x; \alpha, \beta, \lambda) = \sum_{i=0}^{\infty} P(i; \lambda/2) I_x(\alpha+i, \beta)$$ where \(I_x(\alpha, \beta)\) is the incomplete beta function.

Usage

non_central_beta_distribution(alpha, beta, lambda)

non_central_beta_pdf(x, alpha, beta, lambda)

non_central_beta_lpdf(x, alpha, beta, lambda)

non_central_beta_cdf(x, alpha, beta, lambda)

non_central_beta_lcdf(x, alpha, beta, lambda)

non_central_beta_quantile(p, alpha, beta, lambda)

Arguments

alpha

first shape parameter (alpha > 0)

beta

second shape parameter (beta > 0)

lambda

noncentrality parameter (lambda >= 0)

x

quantile (0 <= x <= 1)

p

probability (0 <= p <= 1)

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

# Noncentral Beta distribution with shape parameters alpha = 2, beta = 3
# and noncentrality parameter lambda = 1
dist <- non_central_beta_distribution(2, 3, 1)
# Apply generic functions
cdf(dist, 0.5)
#> [1] 0.5977904
logcdf(dist, 0.5)
#> [1] -0.514515
pdf(dist, 0.5)
#> [1] 1.643543
logpdf(dist, 0.5)
#> [1] 0.4968546
hazard(dist, 0.5)
#> [1] 4.086286
chf(dist, 0.5)
#> [1] 0.910782
mean(dist)
#> [1] 0.44664
median(dist)
#> [1] 0.4416064
mode(dist)
#> [1] 0.4262677
range(dist)
#> [1] 0 1
quantile(dist, 0.2)
#> [1] 0.2549084
standard_deviation(dist)
#> [1] 0.2040433
support(dist)
#> [1] 0 1
variance(dist)
#> [1] 0.04163366

# Convenience functions
non_central_beta_pdf(0.5, 2, 3, 1)
#> [1] 1.643543
non_central_beta_lpdf(0.5, 2, 3, 1)
#> [1] 0.4968546
non_central_beta_cdf(0.5, 2, 3, 1)
#> [1] 0.5977904
non_central_beta_lcdf(0.5, 2, 3, 1)
#> [1] -0.514515
non_central_beta_quantile(0.5, 2, 3, 1)
#> [1] 0.4416064