cfN_Poisson: Characteristic function of the Poisson distribution

In gajdosandrej/CharFunToolR: Numerical Computation Cumulative Distribution Function and Probability Density Function from Characteristic Function

Characteristic function of the Poisson distribution

Description

cfN_Poisson(t, lambda, cfX) evaluates the characteristic function cf(t) of the Poisson distribution with the rate parameter lambda > 0, i.e.

cfN_Poisson(t, \lambda) = exp(\lambda*(exp(1i*t)-1))

. For more details see [4].

cfN_Poisson(t, lambda, cfX) evaluates the compound characteristic function

cf(t) = cfN_Poisson(-1i*log(cfX(t)), \lambda),

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_Poisson(t, lambda = 1, cfX)

Arguments

t	vector or array of real values, where the CF is evaluated.
lambda	rate, lambda > 0, default value lambda = 1.
cfX	function.

Value

Characteristic function cf(t) of the Poisson distribution.

Note

Ver.: 16-Sep-2018 19:02:30 (consistent with Matlab CharFunTool v1.3.0, 15-Nov-2016 13:36:26).

References

[1] WITKOVSKY V., WIMMER G., DUBY T. (2016). Computing the aggregate loss distribution based on numerical inversion of the compound empirical characteristic function of frequency and severity. Preprint submitted to Insurance: Mathematics and Economics.

[2] DUBY T., WIMMER G., WITKOVSKY V.(2016). MATLAB toolbox CRM for computing distributions of collective risk models. Preprint submitted to Journal of Statistical Software.

[3] WITKOVSKY V. (2016). Numerical inversion of a characteristic function: An alternative tool to form the probability distribution of output quantity in linear measurement models. Acta IMEKO, 5(3), 32-44.

[4] WIMMER G., ALTMANN G. (1999). Thesaurus of univariate discrete probability distributions. STAMM Verlag GmbH, Essen, Germany. ISBN 3-87773-025-6.

See Also

For more details see WIKIPEDIA: https://en.wikipedia.org/wiki/Poisson_distribution,
https://en.wikipedia.org/wiki/Compound_Poisson_distribution.

Other Discrete Probability Distribution: cfN_Binomial(), cfN_Delaporte(), cfN_GeneralizedPoisson(), cfN_Geometric(), cfN_Logarithmic(), cfN_NegativeBinomial(), cfN_PolyaEggenberger(), cfN_Quinkert(), cfN_Waring()

Examples

## EXAMPLE1
# CF of the Poisson distribution with the parameter lambda = 10
lambda <- 10
t <- seq(-10, 10, length.out = 501)
plotReIm(function(t)
        cfN_Poisson(t, lambda), t,
        title = "CF of the Poisson distribution with the parameter lambda = 10")

## EXAMPLE2
# CF of the compound Poisson-Exponential distribution
lambda1 <- 10
lambda2 <- 5
cfX <- function(t)
        cfX_Exponential(t, lambda2)
t <- seq(-10, 10, length.out = 501)
plotReIm(function(t)
        cfN_Poisson(t, lambda1, cfX), t,
        title = "CF of the compound Poisson-Exponential distribution")

## EXAMPLE3
# PDF/CDF of the compound Poisson-Exponential distribution
lambda1 <- 10
lambda2 <- 5
cfX <- function(t)
        cfX_Exponential(t, lambda2)
cf <- function(t)
        cfN_Poisson(t, lambda1, cfX)
x <- seq(0, 8, length.out = 101)
prob <- c(0.9, 0.95, 0.99)
options <- list()
options$isCompound <- TRUE
result <- cf2DistGP(cf, x, prob, options)

gajdosandrej/CharFunToolR documentation built on June 3, 2024, 7:46 p.m.