Continuous parameter binomial distribution functions

Density, distribution function, and quantile function for the continuous parameter binomial (CPB), and its zero-truncated form. The CPB has a hard ceiling: King's CPB (the default) puts no probability above \(\lceil \lambda/(1-\alpha) \rceil\), the floor support (support = "floor", the definition before version 0.2.0) none above \(\lfloor \lambda/(1-\alpha) \rfloor\). Complements the simulator rcpb().

Usage

dcpb(
  x,
  lambda,
  alpha,
  truncated = FALSE,
  log = FALSE,
  support = c("ceiling", "floor")
)

pcpb(
  q,
  lambda,
  alpha,
  truncated = FALSE,
  lower.tail = TRUE,
  log.p = FALSE,
  support = c("ceiling", "floor")
)

qcpb(
  p,
  lambda,
  alpha,
  truncated = FALSE,
  lower.tail = TRUE,
  log.p = FALSE,
  support = c("ceiling", "floor")
)

Arguments

x, q

Vector of quantiles (non-negative integers).

lambda

Rate parameter (scalar or vector, recycled). It is the mean where lambda / (1 - alpha) is an integer; otherwise the mean lies above it on King's support and below it on the floor support.

alpha

Shape parameter in (0, 1).

truncated

If TRUE, use the zero-truncated CPB. On the floor support, where the implied ceiling lambda / (1 - alpha) is below 1 the CPB puts all its mass on 0, and its zero-truncated form is the point mass at 1, the zero-truncated distribution for every ceiling in [1, 2); King's support always contains 1.

log, log.p

If TRUE, probabilities are given as log.

support

The CPB's support, "ceiling" (King's CPB, the default) or "floor"; see cpb().

lower.tail

If TRUE (default), probabilities are \(P(X \le x)\).

p

Vector of probabilities.

Value

dcpb a density, pcpb a distribution function, qcpb a quantile.

See also

rcpb()

Examples

dcpb(0:5, lambda = 3, alpha = 0.5)
#> [1] 0.015625 0.093750 0.234375 0.312500 0.234375 0.093750
pcpb(3, lambda = 3, alpha = 0.5)
#> [1] 0.65625
qcpb(0.9, lambda = 3, alpha = 0.5)
#> [1] 5