ZeroTruncatedBinomial {actuar} | R Documentation |

Density function, distribution function, quantile function and random
generation for the Zero-Truncated Binomial distribution with
parameters `size`

and `prob`

.

dztbinom(x, size, prob, log = FALSE) pztbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE) qztbinom(p, size, prob, lower.tail = TRUE, log.p = FALSE) rztbinom(n, size, prob)

`x` |
vector of (strictly positive integer) quantiles. |

`q` |
vector of quantiles. |

`p` |
vector of probabilities. |

`n` |
number of observations. If |

`size` |
number of trials (strictly positive integer). |

`prob` |
probability of success on each trial. |

`log, log.p` |
logical; if |

`lower.tail` |
logical; if |

The zero-truncated binomial distribution with `size`

*= n* and `prob`

*= p* has probability mass function

*
p(x) = choose(n, x) [p^x (1-p)^(n-x)]/[1 - (1-p)^n]*

for *x = 1, …, n* and *0 < p ≤ 1*, and *p(1) = 1*
when *p = 0*.
The cumulative distribution function is

*
P(x) = [F(x) - F(0)]/[1 - F(0)],*

where *F(x)* is the distribution function of the standard binomial.

The mean is *np/(1 - (1-p)^n)* and the variance is
*np[(1-p) - (1-p+np)(1-p)^n]/[1 - (1-p)^n]^2*.

In the terminology of Klugman et al. (2012), the zero-truncated
binomial is a member of the *(a, b, 1)* class of
distributions with *a = -p/(1-p)* and *b = (n+1)p/(1-p)*.

If an element of `x`

is not integer, the result of
`dztbinom`

is zero, with a warning.

The quantile is defined as the smallest value *x* such that
*P(x) ≥ p*, where *P* is the distribution function.

`dztbinom`

gives the probability mass function,
`pztbinom`

gives the distribution function,
`qztbinom`

gives the quantile function, and
`rztbinom`

generates random deviates.

Invalid `size`

or `prob`

will result in return value
`NaN`

, with a warning.

The length of the result is determined by `n`

for
`rztbinom`

, and is the maximum of the lengths of the
numerical arguments for the other functions.

Functions `{d,p,q}ztbinom`

use `{d,p,q}binom`

for all
but the trivial input values and *p(0)*.

`rztbinom`

uses the simple inversion algorithm suggested by
Peter Dalgaard on the r-help mailing list on 1 May 2005
(https://stat.ethz.ch/pipermail/r-help/2005-May/070680.html).

Vincent Goulet vincent.goulet@act.ulaval.ca

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012),
*Loss Models, From Data to Decisions, Fourth Edition*, Wiley.

`dbinom`

for the binomial distribution.

dztbinom(1:5, size = 5, prob = 0.4) dbinom(1:5, 5, 0.4)/pbinom(0, 5, 0.4, lower = FALSE) # same pztbinom(1, 2, prob = 0) # point mass at 1 qztbinom(pztbinom(1:10, 10, 0.6), 10, 0.6) n <- 8; p <- 0.3 x <- 0:n title <- paste("ZT Binomial(", n, ", ", p, ") and Binomial(", n, ", ", p,") PDF", sep = "") plot(x, dztbinom(x, n, p), type = "h", lwd = 2, ylab = "p(x)", main = title) points(x, dbinom(x, n, p), pch = 19, col = "red") legend("topright", c("ZT binomial probabilities", "Binomial probabilities"), col = c("black", "red"), lty = c(1, 0), lwd = 2, pch = c(NA, 19))

[Package *actuar* version 3.1-4 Index]