Description Usage Arguments Value See Also Examples
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).
1 | cfN_GeneralizedPoisson(t, a, p = 0.5, cfX)
|
t |
numerical values (number, vector...) |
a |
variable mean, a > 0 |
p |
success probability, 0 ≤ p ≤ 1, default value p = 1/2 |
cfX |
function |
characteristic function cf(t) of the Poisson distribution with rate lambda
For more details
Other Discrete Probability Distribution: cfN_Binomial
,
cfN_Delaporte
, cfN_Geometric
,
cfN_Logarithmic
,
cfN_NegativeBinomial
,
cfN_Poisson
,
cfN_PolyaEggenberger
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)
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