VGAM (version 0.8-2)

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. If length(n) > 1 then the length is taken to be the number required.

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

(ii = dposbinom(0:size, size, prob))
cumsum(ii) - 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

barplot(rbind(dposbinom(x = 0:size, size, prob),
                        dbinom(x = 0:size, size, prob)),
        beside = TRUE, col = c("blue","green"),
        main=paste("Positive-binomial(", size, ",", prob, ") (blue) vs",
        " Binomial(", size, ",", prob, ") (green)", sep=""),
        names.arg = as.character(0:size), las=1)