Cumulative Distribution Function (CDF) of Zero-Truncated Poisson Distribution

Computes the cumulative distribution function (CDF) for a zero-truncated Poisson (TP) distribution. This function modifies the standard Poisson CDF to account for truncation at zero (i.e., only supports $y > 0$).

Usage

cdf.tp(y, lambda, lower.tail = FALSE, log.p = FALSE)

Arguments

y: Numeric vector of quantiles (count values) for which to compute the CDF.
lambda: Numeric vector or scalar giving the mean parameter ($\lambda$) of the Poisson distribution.
lower.tail: Logical; if TRUE (default), probabilities are $P(Y \le y)$. If FALSE, probabilities are $P(Y > y)$.
log.p: Logical; if TRUE, probabilities $p$ are given as $\log(p)$.

Value

A numeric vector of the same length as the input, containing:

CDF values ($P(Y \le y \mid Y > 0)$) if lower.tail = TRUE.
Upper-tail probabilities ($P(Y > y \mid Y > 0)$) if lower.tail = FALSE.
Log-probabilities if log.p = TRUE.

Details

The function computes probabilities for the zero-truncated version of the Poisson distribution: $$P(Y \le y \mid Y > 0) = \frac{P(Y \le y) - P(Y = 0)}{1 - P(Y = 0)}.$$

Internally, it uses the ppois() function from base R for computing Poisson probabilities, and performs calculations on the log scale for improved numerical stability.

When lower.tail = FALSE, it returns the upper-tail probabilities $P(Y > y \mid Y > 0)$ instead.

Examples