CharFun (version 0.1.0)

cfN_Logarithmic: Characteristic function of Logarithmic distribution

Description

cfN_Logarithmic(t, p) evaluates the characteristic function cf(t) of the Logarithmic distribution defined on non-negative integers n =0, 1, ..., with the parameters p (success probability p in [0,1]), i.e. cfN_Logarithmic(t, p) = log(1 - p * exp(1i*t)) / log(1 - p)

cfN_Logarithmic(t,p,cfX) evaluates the compound characteristic function cf(t) = cfN_Logarithmic(-1i*log(cfX(t)), p), where cfX is function handle of the characteristic function cfX(t) of a continuous distribution and/or random variable X.

Note that such CF is characteristic function of the compound distribution, i.e. distribution of the random variable Y = X_1 + ... + X_N, where X_i ~ F_X are i.i.d. random variables with common CF cfX(t), and N ~ F_N is independent RV with its CF given by cfN(t).

Usage

cfN_Logarithmic(t, p = 0.5, cfX)

Arguments

t

numerical values (number, vector...)

p

success probability, \(0 \le p \le 1\), default value p = 1/2

cfX

function

Value

characteristic function cf(t) of the Logarithmic distribution p success probability

See Also

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

Other Discrete Probability Distribution: cfN_Binomial, cfN_Delaporte, cfN_GeneralizedPoisson, cfN_Geometric, cfN_NegativeBinomial, cfN_Poisson, cfN_PolyaEggenberger

Examples

## EXAMPLE1 (CF of the Logarithmic distribution with the parameter p = 0.5)
p <- 0.5
t <- seq(-10, 10, length.out = 501)
plotGraf(function(t)
  cfN_Logarithmic(t, p), t, title = "CF of the Logarithmic distribution with the parameter p = 0.5")

#' ## EXAMPLE2 (CF of the compound Logarithmic-Exponential distributionn)
p <- 0.5
lambda <- 5
cfX <- function(t)
  cfX_Exponential(t, lambda)
t <- seq(-10, 10, length.out = 501)
plotGraf(function(t)
  cfN_Logarithmic(t, p, cfX), t, title = "CF of the compound Logarithmic-Exponential distribution")

## EXAMPLE3 (PDF/CDF of the compound Logarithmic-Exponential distribution)
p <- 0.5
lambda <- 5
cfX <- function(t)
  cfX_Exponential(t, lambda)
cf <- function(t)
  cfN_Logarithmic(t, p, cfX)
x <- seq(0, 3, length.out = 101)
prob <- c(0.9, 0.95, 0.99)
result <- cf2DistGP(cf, x, prob, isCompound = TRUE)