Description Usage Arguments Value See Also Examples
cfN_Poisson(t, lambda) 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))
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).
1 | cfN_Poisson(t, lambda = 1, cfX)
|
t |
numerical values (number, vector...) |
lambda |
rate, lambda > 0, default value lambda = 1 |
cfX |
function |
characteristic function cf(t) of the Poisson distribution with rate lambda
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
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 | ## EXAMPLE1 (CF of the Poisson distribution with the parameter lambda = 10)
lambda <- 10
t <- seq(-10, 10, length.out = 501)
plotGraf(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 distributionn)
lambda1 <- 10
lambda2 <- 5
cfX <- function(t)
cfX_Exponential(t, lambda2)
t <- seq(-10, 10, length.out = 501)
plotGraf(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)
result <- cf2DistGP(cf, x, prob, isCompound = TRUE)
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