CharFun (version 0.1.0)

cfN_NegativeBinomial: Characteristic function of Negative-Binomial distribution

Description

cfN_NegativeBinomial(t, r, p) evaluates the characteristic function cf(t) of the Negative-Binomial distribution with the parameters r (number of failures until the experiment is stopped, r in N) and p (success probability in each experiment, p in [0,1]), i.e. cfN_NegativeBinomial(t, r, p) = p^r * (1 - (1-p) * e^(1i*t))^(-r)

cfN_NegativeBinomial(t, r, p, cfX) evaluates the compound characteristic function cf(t) = cfN_NegativeBinomial(-1i*log(cfX(t)), r, 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_NegativeBinomial(t, r = 10, p = 0.5, cfX)

Arguments

t

numerical values (number, vector...)

r

number of trials

p

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

cfX

function

Value

characteristic function cf(t) of the Negative-Binomial distribution with n trials and p success probability

See Also

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

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

Examples

## EXAMPLE1 (CF of the Negative Binomial distribution with r = 5, p = 0.3)
r <- 5
p <- 0.3
t <- seq(-15, 15, length.out = 1001)
plotGraf(function(t)
  cfN_NegativeBinomial(t, r, p), t,
  title = "CF of the Negative Binomial distribution with r = 5, p = 0.3")

## EXAMPLE2 (PDF/CDF of the compound NegativeBinomial-Exponential distribution)
r <- 5
p <- 0.3
lambda <- 5
cfX <- function(t)
  cfX_Exponential(t, lambda)
cf <- function(t)
  cfN_NegativeBinomial(t, r, p, cfX)
x <- seq(0, 10, length.out = 101)
prob <- c(0.9, 0.95, 0.99)
result <- cf2DistGP(cf, x, prob, isCompound = TRUE)