CharFun (version 0.1.0)

cfX_PearsonV: Characteristic function of Pearson type V distribution

Description

cfX_PearsonV(t, alpha, beta) evaluates the characteristic function cf(t) of the Pearson type V distribution with the parameters alpha (shape, alpha > 0) and beta (scale, beta > 0), computed for real vector argument t, i.e.

cfX_PearsonV(t, alpha, beta) = (2/gamma(alpha)) * (-1i*t/beta)^(alpha/2) * besselk(alpha,2*sqrt(-1i*t/beta)), where besselk(a,z) denotes the modified Bessel function of the second.

Usage

cfX_PearsonV(t, alpha = 1, beta = 1)

Arguments

t

numerical real values (number, vector...)

alpha

shape, alpha > 0, default value alpha = 1

beta

scale > 0, default value beta = 1

Value

characteristic function cf(t) of the Gamma distribution

See Also

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

Other Continuous Probability distribution: cfS_Arcsine, cfS_Beta, cfS_Gaussian, cfS_Rectangular, cfS_StudentT, cfS_Trapezoidal, cfS_Triangular, cfX_Beta, cfX_ChiSquared, cfX_Exponential, cfX_Gamma, cfX_InverseGamma, cfX_LogNormal, cfX_Normal, cfX_Rectangular, cfX_Triangular

Examples

## EXAMPLE1 ((CF of the PearsonV distribution)
alpha <- 3 / 2
beta <- 2 / 3
t <- seq(-10, 10, length.out = 1001)
plotGraf(function(t)
  cfX_PearsonV(t, alpha, beta), t,
  title = "CF of the PearsonV distribution with alpha = 3/2, beta = 2/3")

## EXAMPLE2 (PDF/CDF of the Beta distribution with alpha = 3/2, beta = 2/3)
alpha <- 3 / 2
beta <- 2 / 3
prob <- c(0.9, 0.95, 0.99)
x <- seq(0, 40, length.out = 101)
cf <- function(t)
  cfX_PearsonV(t, alpha, beta)
result <-
  cf2DistGP(cf,
            x,
            prob,
            xMin = 0,
            N = 2 ^ 10,
            SixSigmaRule = 10)

## EXAMPLE3 (PDF/CDF of the compound Binomial-PearsonV distribution)
n <- 25
p <- 0.3
alpha <- 3 / 2
beta <- 2 / 3
prob <- c(0.9, 0.95, 0.99)
x <- seq(0, 200, length.out = 101)
cfX <- function(t)
  cfX_PearsonV(t, alpha, beta)
cf <- function(t)
  cfN_Binomial(t, n, p, cfX)

result <- cf2DistGP(cf, x, prob, isCompound = TRUE, N = 2 ^ 10)