cf_Laplace: Characteristic function of a linear combination (resp....

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

Characteristic function of a linear combination (resp. convolution) of independent LAPLACE random variables with location parameter mu (real) and scale parameter beta > 0.

Description

cf_Laplace(t,mu,beta,coef,niid) evaluates the characteristic function cf(t) of Y = \sum_{i=1}^N coef_i * X_i, where X_i ~ Laplace (\mu_i,\beta_{i}) are inedependent RVs, with real location parameters \mu_{i} and the scale parameters \beta_{i} > 0, for i = 1,...,N.

The characteristic function of Y is defined by

cf(t)=Prod(exp(li* t * coef(i) \eqn {mu(i)} ) / (1+(t*coef(i)*\eqn{beta(i)})^2) )

Usage

cf_Laplace(t, mu, beta, coef, niid)

Arguments

t	vector or array of real values, where the CF is evaluated.
mu	vector of real location parameters. If empty, default value is mu = 0.
beta	vector of the scale parameters beta > 0. If empty, default value is beta = 1.
coef	vector of the coefficients of the linear combination of the LAPLACE 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) * 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 LAPLACE random variables.

Note

Ver.: 08-Aug-2021 16:19:30 (consistent with Matlab CharFunTool v1.5.1, 16-Aug-2018 16:00:43).

See Also

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

## EXAMPLE1
# CF of the Laplace RV
mu <- 0
beta <- 1
t  <- seq(from = -10,
              to = 10,
              length.out =201)
plotReIm(function(t)
        cf_Laplace(t, mu, beta),
        t,
        title = "Characteristic function of the Laplace RVs")



##EXAMPLE2
# PDF/CDF of the Laplace RV
mu <- 0
beta <- 1
x <- seq(-5, 5, length.out = 101)

prop <- c(0.80, 0.85, 0.90, 0.925, 0.95, 0.975, 0.99, 0.995, 0.999)
cf <- function(t)
        cf_Laplace(t, mu, beta)
result <- cf2DistGP(cf, x, prob)

##EXAMPLE3
# PDF/CDF of the linear combination of Laplace RVs
mu <- c(-4, -1, 2, 3)
beta <- c(0.1, 0.2, 0.3, 0.4)
coef <- c(1, 2, 3, 4)
prob <- c(0.80, 0.85, 0.90, 0.925, 0.95, 0.975, 0.99, 0.995, 0.999)
cf <- function(t)
        cf_Laplace(t, mu, beta, coef)
options <- list()
options$N <- 2^12
result <- cf2DistGP(cf,prob=prob,options=options)

## EXAMPLE 4
# PDF/CDF of the linear combination of the Laplace RVs
mu <- c(-10, 10, 20, 30, 40)
beta <- c(1, 2, 3, 4, 5)
coef <- c(1/2, 1, 3/4, 5, 1)
prop <- c(0.80, 0.85, 0.90, 0.925, 0.95, 0.975, 0.99, 0.995, 0.999)
cf <- function(t)
        cf_Laplace(t, mu, beta, coef)
options <- list()
options$N <- 2^12
result <- cf2DistGP(cf,c(),prob,options)

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