ZeroTruncatedPoisson {actuar} | R Documentation |

Density function, distribution function, quantile function, random
generation for the Zero-Truncated Poisson distribution with parameter
`lambda`

.

dztpois(x, lambda, log = FALSE) pztpois(q, lambda, lower.tail = TRUE, log.p = FALSE) qztpois(p, lambda, lower.tail = TRUE, log.p = FALSE) rztpois(n, lambda)

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

`q` |
vector of quantiles. |

`p` |
vector of probabilities. |

`n` |
number of values to return. |

`lambda` |
vector of (non negative) means. |

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

`lower.tail` |
logical; if |

The zero-truncated Poisson distribution has probability mass function

*
p(x) = lambda^x exp(-lambda)/[x! (1 - exp(-lambda))]
= lambda^x/[x! (e^lambda - 1)]*

for *x = 1, 2, ...*, and *p(1) = 1* when *λ = 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 Poisson.

The mean is *λ/(1 -
exp(-λ))* and the variance is
*
λ[1 - (λ+1)exp(-λ)]/(1 - exp(-λ))^2*.

In the terminology of Klugman et al. (2012), the zero-truncated
Poisson is a member of the *(a, b, 1)* class of distributions
with *a = 0* and *b = λ*.

If an element of `x`

is not integer, the result of
`dztpois`

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.

`dztpois`

gives the (log) probability mass function,
`pztpois`

gives the (log) distribution function,
`qztpois`

gives the quantile function, and
`rztpois`

generates random deviates.

Invalid `lambda`

will result in return value `NaN`

, with a
warning.

The length of the result is determined by `n`

for
`rztpois`

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

Functions `{d,p,q}ztpois`

use `{d,p,q}pois`

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

`rztpois`

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.

`dpois`

for the standard Poisson distribution.

dztpois(1:5, lambda = 1) dpois(1:5, lambda = 1)/ppois(0, 1, lower = FALSE) # same pztpois(1, lambda = 0) # point mass at 1 qztpois(pztpois(1:10, 1), 1) x <- seq(0, 8) plot(x, dztpois(x, 2), type = "h", lwd = 2, ylab = "p(x)", main = "Zero-Truncated Poisson(2) and Poisson(2) PDF") points(x, dpois(x, 2), pch = 19, col = "red") legend("topright", c("ZT Poisson probabilities", "Poisson probabilities"), col = c("black", "red"), lty = c(1, 0), lwd = 2, pch = c(NA, 19))

[Package *actuar* version 3.1-4 Index]