# cf_TrapezoidalSymmetric: Characteristic function of a linear combination of... In gajdosandrej/CharFunToolR: Numerical Computation Cumulative Distribution Function and Probability Density Function from Characteristic Function

 cf_TrapezoidalSymmetric R Documentation

## Characteristic function of a linear combination of independent zero-mean symmetric TRAPEZOIDAL random variables

### Description

cf_TrapezoidalSymmetric(t, lambda, coef, niid) evaluates the characteristic function of a linear combination (resp. convolution) of independent zero-mean symmetric TRAPEZOIDAL random variables defined on the interval (-1,1).

That is, cf_TrapezoidalSymmetric evaluates the characteristic function cf(t) of Y = sum_{i=1}^N coef_i * X_i, where X_i ~ TrapezoidalSymmetric(\lambda_i) are independent RVs defined on (-1,1), for all i = 1,...,N.

The characteristic function of X ~ TrapezoidalSymmetric(\lambda), with E(X) = 0 and Var(X) = (1+\lambda^2)/6, is

cf(t) = cf_TrapezoidalSymmetric(t) = cf_RectangularSymmetric(w*t))*cf_RectangularSymmetric((1-w)*t) = (sin(w*t)/(w*t))*(sin((1-w)*t)/((1-w)*t)),

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

### Usage

cf_TrapezoidalSymmetric(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 the coefficients of the linear combination of the zero-mean symmetric TRAPEZOIDAL 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 niid, such that Z = Y + ... + Y is sum of niid iid random variables Y, where each Y = sum_{i=1}^N coef(i) * log(X_i) is independently and identically distributed random variable. If empty, default value is niid = 1.

### Value

Characteristic function cf(t) of a linear combination of independent zero-mean symmetric TRAPEZOIDAL random variables.

### Note

Ver.: 16-Sep-2018 18:39:29 (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.

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_Trapezoidal(), 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_TriangularSymmetric(), cf_vonMises()

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

### Examples

## EXAMPLE 1
#CF of the symmetric Trapezoidal distribution with lambda = 1/2
lambda <- 1 / 2
t <- seq(from = -50,
to = 50,
length.out = 201)
plotReIm(function(t)
cf_TrapezoidalSymmetric(t, lambda), t,
title = "CF of the symmetric Trapezoidal distribution on (-1,1)")

## EXAMPLE 2
# CF of a linear combination of independent Trapezoidal RVs
t <- seq(from = -20,
to = 20,
length.out = 201)
lambda <- c(3, 3, 4, 4, 5) / 7
coef <- c(1, 2, 3, 4, 5) / 15
plotReIm(function(t)
cf_TrapezoidalSymmetric(t, lambda, coef), t,
title = "CF of a linear combination of independent Trapezoidal RVs")

## EXAMPLE 3
# PDF/CDF of a weighted linear combination of independent Trapezoidal RVs
lambda <- c(3, 3, 4, 4, 5) / 7
coef <- c(1, 2, 3, 4, 5) / 15
cf <- function(t)
cf_TrapezoidalSymmetric(t, lambda, coef)
x <- seq(from = -1,
to = 1,
length.out = 201)
prob <- c(0.9, 0.95, 0.99)
options <- list()
options$N <- 2 ^ 12 options$xMin <- -1
options\$xMax <- 1
result <- cf2DistGP(cf, x, prob, options)


gajdosandrej/CharFunToolR documentation built on Aug. 22, 2023, 9:58 a.m.