VGAM (version 0.7-3)

Posbinom: Positive-Binomial Distribution

Description

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

Usage

dposbinom(x, size, prob, log = FALSE)
pposbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE)
qposbinom(p, size, prob, lower.tail = TRUE, log.p = FALSE)
rposbinom(n, size, prob)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. Must be a single positive integer.

size

number of trials. It is the $N$ symbol in the formula given in posbinomial.

prob

probability of success on each trial.

log, log.p, lower.tail

Arguments that are passed on to pbinom etc.

Value

• dposbinom gives the density, pposbinom gives the distribution function, qposbinom gives the quantile function, and rposbinom generates random deviates.

Details

The positive-binomial distribution is a binomial distribution but with the probability of a zero being zero. The other probabilities are scaled to add to unity. The mean therefore is $$\mu / (1-(1-\mu)^N)$$ where $\mu$ is the argument prob above. As $\mu$ increases, the positive-binomial and binomial distributions become more similar. Unlike similar functions for the binomial distribution, a zero value of prob is not permitted here.

See Also

posbinomial, rbinom.

Examples

prob = 0.2
size = 10
y = rposbinom(n=1000, size, prob)
table(y)
mean(y)  # Sample mean
prob / (1-(1-prob)^size) # Population mean

(i = dposbinom(0:size, size, prob))
cumsum(i) - pposbinom(0:size, size, prob)  # Should be 0s
table(rposbinom(100, size, prob))

table(qposbinom(runif(1000), size, prob))
round(dposbinom(1:10, size, prob) * 1000) # Should be similar

x = 0:size
plot(x, dposbinom(x, size, prob), type="h", ylab="Probability",
     main=paste("Positive-binomial(", size, ",", prob, ") (blue) vs",
     " Binomial(", size, ",", prob, ") (red and shifted slightly)", sep=""),
     lwd=2, col="blue", las=1)
lines(x+0.05, dbinom(x, size, prob), type="h", lwd=2, col="red")