VGAM (version 1.1-3)

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)

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.

Value

dpospois gives the density, ppospois gives the distribution function, qpospois gives the quantile function, and rpospois generates random deviates.

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

Gaitpois, pospoisson, zapoisson, zipoisson, rpois.

Examples

# NOT RUN {
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

# }
# NOT RUN {
 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) 
# }