VGAM (version 0.7-1)

Pospois: Positive-Poisson Distribution

Description

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

Usage

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

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. Must be a single positive integer.

lambda

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

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 is not permitted here.

See Also

pospoisson, zapoisson, rpois.

Examples

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

(i = dpospois(0:7, lambda))
cumsum(i) - 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

x = 0:7
plot(x, dpospois(x, lambda), type="h", ylab="Probability",
     main=paste("Positive Poisson(", lambda, ") (blue) vs",
     " Poisson(", lambda, ") (red & shifted slightly)", sep=""),
     lwd=2, col="blue", las=1)
lines(x+0.05, dpois(x, lambda), type="h", lwd=2, col="red")