cfX_Gamma: Characteristic function of the GAMMA distribution

In gajdosandrej/CharFunToolR: Numerical Computation Cumulative Distribution Function and Probability Density Function from Characteristic Function

Characteristic function of the GAMMA distribution

Description

cfX_Gamma(t, alpha, beta, coef, niid) evaluates the characteristic function cf(t) of the GAMMA distribution with the shape parameter alpha > 0 and the rate parameter beta > 0.

cfX_Gamma is an ALIAS NAME of the more general function cf_Gamma, used to evaluate the characteristic function of a linear combination of independent GAMMA distributed random variables.

The characteristic function of the GAMMA distribution is defined by

cf(t) = (1 - i*t/\beta)^(-\alpha).

Usage

cfX_Gamma(t, alpha, beta, coef, niid)

Arguments

t	vector or array of real values, where the CF is evaluated.
alpha	the shape parameter alpha > 0. If empty, default value is alpha = 1.
beta	the rate (1/scale) parameter beta > 0. If empty, default value is beta = 1.
coef	vector of coefficients of the linear combination of Gamma distributed random variables. If coef is scalar, it is assumed that all coefficients are equal. If empty, default value is coef = 1.
niid	scalar convolution coeficient.

Value

Characteristic function cf(t) of the GAMMA distribution.

Note

Ver.: 16-Sep-2018 19:22:15 (consistent with Matlab CharFunTool v1.3.0, 24-Jun-2017 10:07:43).

See Also

For more details see WIKIPEDIA: https://en.wikipedia.org/wiki/Gamma_distribution.

Other Continuous Probability Distribution: cfS_Arcsine(), cfS_Beta(), cfS_Gaussian(), cfS_Laplace(), cfS_Rectangular(), cfS_Student(), cfS_TSP(), cfS_Trapezoidal(), cfS_Triangular(), cfS_Wigner(), cfX_ChiSquare(), cfX_Exponential(), cfX_FisherSnedecor(), cfX_InverseGamma(), cfX_LogNormal(), cf_ArcsineSymmetric(), cf_BetaNC(), cf_BetaSymmetric(), cf_Beta(), cf_ChiSquare(), cf_Exponential(), cf_FisherSnedecorNC(), cf_FisherSnedecor(), cf_Gamma(), cf_InverseGamma(), cf_Laplace(), cf_LogRV_BetaNC(), cf_LogRV_Beta(), cf_LogRV_ChiSquareNC(), cf_LogRV_ChiSquare(), cf_LogRV_FisherSnedecorNC(), cf_LogRV_FisherSnedecor(), cf_LogRV_MeansRatioW(), cf_LogRV_MeansRatio(), cf_LogRV_WilksLambdaNC(), cf_LogRV_WilksLambda(), cf_Normal(), cf_RectangularSymmetric(), cf_Student(), cf_TSPSymmetric(), cf_TrapezoidalSymmetric(), cf_TriangularSymmetric(), cf_vonMises()

Examples

## EXAMPLE 1
# CF of the Gamma distribution with alpha = 2, beta = 2
alpha <- 2
beta <- 2
t <- seq(-20, 20, length.out = 501)
plotReIm(function(t)
        cfX_Gamma(t, alpha, beta),
        t,
        title = "CF of the Gamma distribution with alpha = 2, beta = 2")

## EXAMPLE 2
# PDF/CDF of the Gamma distribution with alpha = 2, beta = 2
alpha <- 2
beta <- 2
x <- seq(from = 0,
         to = 5,
         length.out = 101)
prob <- c(0.9, 0.95, 0.99)
options <- list()
options$xMin <- 0
options$N <- 2 ^ 14
cf <- function(t)
        cfX_Gamma(t, alpha, beta)
result <- cf2DistGP(cf, x, prob, options)

## EXAMPLE 3
# PDF/CDF of the compound Binomial-Gamma distribution
n = 25
p = 0.3
alpha <- 2
beta <- 2
cfX <- function(t)
        cfX_Gamma(t, alpha, beta)
cf <- function(t)
        cfN_Binomial(t, n, p, cfX)
x <- seq(from = 0,
         to = 25,
         length.out = 101)
prob <- c(0.9, 0.95, 0.99)
options <- list()
options$isCompound <- TRUE
result <- cf2DistGP(cf, x, prob, options)

gajdosandrej/CharFunToolR documentation built on June 3, 2024, 7:46 p.m.