VGAMdata (version 1.1-9)

Pospois: Positive-Poisson Distribution

Description

Density, distribution function, quantile function and random generation for the positive-Poisson distribution.

Usage

dpospois(x, lambda, log = FALSE)
ppospois(q, lambda)
qpospois(p, lambda)
rpospois(n, lambda)

Value

dpospois gives the density,

ppospois gives the distribution function,

qpospois gives the quantile function, and

rpospois generates random deviates.

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. Fed into runif.

lambda

vector of positive means (of an ordinary Poisson distribution). Short vectors are recycled.

log

logical.

Author

T. W. Yee

Details

The positive-Poisson distribution is a Poisson distribution but with the probability of a zero being zero. The other probabilities are scaled to add to unity. The mean therefore is $$\lambda / (1-\exp(-\lambda)).$$ As \(\lambda\) increases, the positive-Poisson and Poisson distributions become more similar. Unlike similar functions for the Poisson distribution, a zero value of lambda returns a NaN.

See Also

Gaitdpois, pospoisson, zapoisson, zipoisson, rpois.

Examples

lambda <- 2; y = rpospois(n = 1000, lambda)
table(y)
mean(y)  # Sample mean
lambda / (1 - exp(-lambda))  # Population mean

(ii <- dpospois(0:7, lambda))
cumsum(ii) - ppospois(0:7, lambda)  # Should be 0s
table(rpospois(100, lambda))

table(qpospois(runif(1000), lambda))
round(dpospois(1:10, lambda) * 1000)  # Should be similar

if (FALSE)  x <- 0:7
barplot(rbind(dpospois(x, lambda), dpois(x, lambda)),
        beside = TRUE, col = c("blue", "orange"),
        main = paste("Positive Poisson(", lambda, ") (blue) vs",
        " Poisson(", lambda, ") (orange)", sep = ""),
        names.arg = as.character(x), las = 1, lwd = 2)