cfN_GeneralizedPoisson: Characteristic function of Generalized Poisson distribution

Description Usage Arguments Value See Also Examples

Description

cfN_GeneralizedPoisson(t, a, p) evaluates the characteristic function cf(t) of the Generalized Poisson with the parameter a (variable mean, a > 0), and p (success probability, 0 ≤ p ≤ 1) i.e. cfN_GeneralizedPoisson(t, a, p) = exp(a*(sum_j=1^Inf ((p*j)^(j-1)*e^(-p*j)/j!)*e^(1i*t*j)-1))

The Generalized-Poisson distribution is equivalent with the Borel-Tanner distribution with parameters (p,m)

cfN_GeneralizedPoisson(t, a, p, cfX) evaluates the compound characteristic function cf(t) = cfN_GeneralizedPoisson(-1i*log(cfX(t)), a, 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

1
cfN_GeneralizedPoisson(t, a, p = 0.5, cfX)

Arguments

t

numerical values (number, vector...)

a

variable mean, a > 0

p

success probability, 0 ≤ p ≤ 1, default value p = 1/2

cfX

function

Value

characteristic function cf(t) of the Poisson distribution with rate lambda

See Also

For more details

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

Examples

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
## EXAMPLE1 (CF of the Generalized-Poisson distribution with a = 10, p = 0.5)
a <- 10
p <- 0.5
t <- seq(-10, 10, length.out = 501)
plotGraf(function(t)
  cfN_GeneralizedPoisson(t, a, p), t,
  title = "CF of the Generalized-Poisson distribution with a = 10, p = 0.5")

## EXAMPLE2 (CF of the compound Generalized-Poisson-Exponential distribution)
a <- 10
p <- 0.5
lambda <- 5
cfX <- function(t)
  cfX_Exponential(t, lambda)
t <- seq(-10, 10, length.out = 501)
plotGraf(function(t)
  cfN_GeneralizedPoisson(t, a, p, cfX), t,
  title = "CF of the compound Generalized-Poisson-Exponential distribution")

## EXAMPLE3 (PDF/CDF of the compound Generalized-Poisson-Exponential distribution)
a <- 10
p <- 0.5
lambda <- 5
cfX <- function(t)
  cfX_Exponential(t, lambda)
cf <- function(t)
  cfN_GeneralizedPoisson(t, a, p, cfX)
x <- seq(0, 15, length.out = 101)
prob <- c(0.9, 0.95, 0.99)
result <- cf2DistGP(cf, x, prob, isCompound = TRUE)

Simkova/CharFun documentation built on May 9, 2019, 1:30 p.m.