cfS_Trapezoidal: Characteristic function of the zero-mean symmetric...

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

Characteristic function of the zero-mean symmetric TRAPEZOIDAL distribution

Description

cfS_Trapezoidal(t, lambda, coef, niid) evaluates the characteristic function of the zero-mean symmetric TRAPEZOIDAL distribution defined on the interval (-1,1).

cfS_Trapezoidal is an ALIAS of the more general function cf_TrapezoidalSymmetric, used to evaluate the characteristic function of a linear combination of independent TRAPEZOIDAL distributed random variables.

The characteristic function of X ~ TrapezoidalSymmetric(\lambda), where 0\le \lambda \le 1 is the offset parameter is defined by

cf(t) = (sin(w*t)/(w*t))*(sin((1-w)*t)/((1-w)*t))

, where w = (1+\lambda)/2.

Usage

cfS_Trapezoidal(t, lambda, coef, niid)

Arguments

t	vector or array of real values, where the CF is evaluated.
lambda	parameter of the offset, 0 \le lambda \le 1. If empty, default value is lambda = 0.
coef	vector of coefficients of the linear combination of Trapezoidal 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 zero-mean symmetric TRAPEZOIDAL distribution.

Note

Ver.: 16-Sep-2018 19:10:20 (consistent with Matlab CharFunTool v1.3.0, 02-Jun-2017 12:08:24).

References

WITKOVSKY V. (2016). Numerical inversion of a characteristic function: An alternative tool to form the probability distribution of output quantity in linear measurement models. Acta IMEKO, 5(3), 32-44.

See Also

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

Other Continuous Probability Distribution: cfS_Arcsine(), cfS_Beta(), cfS_Gaussian(), cfS_Laplace(), cfS_Rectangular(), cfS_Student(), cfS_TSP(), cfS_Triangular(), cfS_Wigner(), cfX_ChiSquare(), cfX_Exponential(), cfX_FisherSnedecor(), cfX_Gamma(), 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()

Other Symmetric Probability Distribution: cfS_Arcsine(), cfS_Beta(), cfS_Gaussian(), cfS_Laplace(), cfS_Rectangular(), cfS_Student(), cf_ArcsineSymmetric(), cf_BetaSymmetric(), cf_RectangularSymmetric(), cf_TSPSymmetric(), cf_TrapezoidalSymmetric()

Examples

## EXAMPLE1
# CF of the symmetric Trapezoidal distribution, lambda = 0.5
lambda <- 0.5
t <- seq(-50, 50, length.out = 501)
plotReIm(function(t)
        cfS_Trapezoidal(t, lambda), t,
        title = "CF of the symmetric Trapezoidal distribution with lambda = 0.5")

## EXAMPLE2
# PDF/CDF of the symmetric Trapezoidal distribution, lambda = 0.5
lambda <- 0.5
cf <- function(t)
        cfS_Trapezoidal(t, lambda)
x <- seq(-1, 1, length.out = 100)
xRange <- 2
options <- list()
options$N <- 2 ^ 10
options$dx <- 2 / pi / xRange
result <- cf2DistGP(cf = cf, x = x, options = options)

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