cfN_Geometric: Characteristic function of the Geometric distribution
View source: R/cfN_Geometric.R
| cfN_Geometric | R Documentation |
Characteristic function of the Geometric distribution
Description
cfN_Geometric(t, p, type, cfX) evaluates the characteristic function cf(t) of the
Geometric distribution.
The standard Geometric distribution (type = "standard" or "zero") is
defined on non-negative integers k = 0,1, \ldots .
The shifted Geometric distribution (type = "shifted")
is defined on positive integers k = 1,2, \ldots .
Both types are parametrized by the success probability parameter p in [0,1]), i.e.
cfN_Geometric(t, p, "standard") = p / (1 - (1-p) * exp(1i*t)),
cfN_Geometric(t, p, "shifted") = exp(1i*t) * (p / (1 - (1-p) * exp(1i*t))).
For more details see [4].
cfN_Geometric(t, p, type, cfX) evaluates the compound characteristic function
cf(t) = Geometric(-1i*log(cfX(t)), 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_Geometric(t, p = 1, type = "standard", cfX)
Arguments
t |
vector or array of real values, where the CF is evaluated. |
p |
success probability, |
type |
|
cfX |
function. |
Value
Characteristic function cf(t) of the Geometric distribution.
Note
Ver.: 16-Sep-2018 19:00:04 (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/Geometric_distribution.
Other Discrete Probability Distribution:
cfN_Binomial(),
cfN_Delaporte(),
cfN_GeneralizedPoisson(),
cfN_Logarithmic(),
cfN_NegativeBinomial(),
cfN_Poisson(),
cfN_PolyaEggenberger(),
cfN_Quinkert(),
cfN_Waring()
Examples
## EXAMPLE1
# CF of the Geometric distribution with the parameter p = 0.5
p <- 0.5
t <- seq(-10, 10, length.out = 501)
plotReIm(function(t)
cfN_Geometric(t, p), t,
title = "CF of the Geometric distribution with the parameter p = 0.5")
## EXAMPLE2
# CF of the Geometric distribution with the parameter p = 0.5, type = "shifted"
p <- 0.5
t <- seq(-10, 10, length.out = 501)
plotReIm(function(t)
cfN_Geometric(t, p, "shifted"), t,
title = "CF of the Geometric distribution with the parameter p = 0.5")
## EXAMPLE3
# CF of the compound Geometric-Exponential distribution
p <- 0.5
lambda <- 5
cfX <- function(t)
cfX_Exponential(t, lambda)
t <- seq(-10, 10, length.out = 501)
plotReIm(function(t)
cfN_Geometric(t, p, 1, cfX), t,
title = "CF of the compound Geometric-Exponential distribution")
## EXAMPLE4
# PDF/CDF of the compound Geometric-Exponential distribution
p <- 0.5
lambda <- 5
cfX <- function(t)
cfX_Exponential(t, lambda)
cf <- function(t)
cfN_Geometric(t, p, cfX = cfX)
x <- seq(0, 4, length.out = 101)
prob <- c(0.9, 0.95, 0.99)
options <- list()
options$isCompound <- TRUE
result <- cf2DistGP(cf, x, prob, options)
