Description Usage Arguments Value See Also Examples
cfN_NegativeBinomial(t, r, p) evaluates the characteristic function cf(t) of the Negative-Binomial distribution with the parameters r (number of failures until the experiment is stopped, r in N) and p (success probability in each experiment, p in [0,1]), i.e. cfN_NegativeBinomial(t, r, p) = p^r * (1 - (1-p) * e^(1i*t))^(-r)
cfN_NegativeBinomial(t, r, p, cfX) evaluates the compound characteristic function cf(t) = cfN_NegativeBinomial(-1i*log(cfX(t)), r, 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_NegativeBinomial(t, r = 10, p = 0.5, cfX)
|
t |
numerical values (number, vector...) |
r |
number of trials |
p |
success probability, 0 ≤ p ≤ 1, default value p = 1/2 |
cfX |
function |
characteristic function cf(t) of the Negative-Binomial distribution with n trials and p success probability
For more details see WIKIPEDIA: https://en.wikipedia.org/wiki/Negative_binomial_distribution
Other Discrete Probability Distribution: cfN_Binomial
,
cfN_Delaporte
,
cfN_GeneralizedPoisson
,
cfN_Geometric
,
cfN_Logarithmic
, cfN_Poisson
,
cfN_PolyaEggenberger
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 | ## EXAMPLE1 (CF of the Negative Binomial distribution with r = 5, p = 0.3)
r <- 5
p <- 0.3
t <- seq(-15, 15, length.out = 1001)
plotGraf(function(t)
cfN_NegativeBinomial(t, r, p), t,
title = "CF of the Negative Binomial distribution with r = 5, p = 0.3")
## EXAMPLE2 (PDF/CDF of the compound NegativeBinomial-Exponential distribution)
r <- 5
p <- 0.3
lambda <- 5
cfX <- function(t)
cfX_Exponential(t, lambda)
cf <- function(t)
cfN_NegativeBinomial(t, r, p, cfX)
x <- seq(0, 10, length.out = 101)
prob <- c(0.9, 0.95, 0.99)
result <- cf2DistGP(cf, x, prob, isCompound = TRUE)
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