cbinom {cbinom} | R Documentation |

Density, distribution function, quantile function and random generation for
a continuous analog to the binomial distribution with parameters `size`

and `prob`

. The usage and help pages are modeled on the d-p-q-r families of
functions for the commonly-used distributions (e.g., `dbinom`

)
in the `stats`

package.

Heuristically speaking, this distribution spreads the standard probability mass
(`dbinom`

) at integer `x`

to the interval
`[x, x + 1]`

in a continuous manner. As a result, the distribution looks
like a smoothed version of the standard, discrete binomial but shifted slightly
to the right. The support of the continuous binomial is `[0, size + 1]`

,
and the mean is approximately `size * prob + 1/2`

.

dcbinom(x, size, prob, log = FALSE) pcbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE) qcbinom(p, size, prob, lower.tail = TRUE, log.p = FALSE) rcbinom(n, size, prob)

`x, q` |
vector of quantiles. |

`p` |
vector of probabilities. |

`n` |
number of observations. If |

`size` |
the |

`prob` |
the |

`log, log.p` |
logical; if TRUE, probabilities p are given as log(p) |

`lower.tail` |
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x] |

The `cbinom`

package is an implementation of Ilienko's (2013) continuous
binomial distribution.

The continuous binomial distribution with `size =`

*N* and
`prob =`

*p* has cumulative distribution function

*
F(x) = B_p(x, N - x + 1)/B(x, N - x + 1)*

for `x`

in `[0, N + 1]`

, where

*
B_p(x, N - x + 1) = integral_0^p (t^(x-1)(1-t)^(y-1))dt*

is the incomplete beta function and

*
B(x, N - x + 1) = integral_0^1 t^(x-1)(1-t)^(y-1)dt*

is the beta function (or
`beta(x, N - x + 1)`

in R). The CDF can be expressed in R as
F(x) = `1 - pbeta(prob, x, size - x + 1)`

and the mean calculated as
`integrate(function(x) pbeta(prob, x, size - x + 1), lower = 0, upper = size + 1)`

.

If an element of `x`

is not in `[0, N + 1]`

, the result of
`dcbinom`

is zero. The PDF `dcbinom(x, size, prob)`

is computed via
numerical differentiation of the CDF = `1 - pbeta(prob, x, size - x + 1)`

.

`dcbinom`

is the density, `pcbinom`

is the distribution function,
`qcbinom`

is the quantile function, and `rcbinom`

generates random
deviates.

The length of the result is determined by `n`

for `rbinom`

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

The numerical arguments other than `n`

are recycled to the length of the
result.

Ilienko, Andreii (2013). Continuous counterparts of Poisson and binomial distributions and their properties. Annales Univ. Sci. Budapest., Sect. Comp. 39: 137-147. http://ac.inf.elte.hu/Vol_039_2013/137_39.pdf

require(graphics) # Compare continous binomial to a standard binomial size <- 20 prob <- 0.2 x <- 0:20 xx <- seq(0, 21, length = 200) plot(x, pbinom(x, size, prob), xlab = "x", ylab = "P(X <= x)") lines(xx, pcbinom(xx, size, prob)) legend('bottomright', legend = c("standard binomial", "continuous binomial"), pch = c(1, NA), lty = c(NA, 1)) mtext(side = 3, line = 1.5, text = "pcbinom resembles pbinom but continuous and shifted") pbinom(x, size, prob) - pcbinom(x + 1, size, prob) # Use "log = TRUE" for more accuracy in the tails and an extended range: n <- 1000 k <- seq(0, n, by = 20) cbind(exp(dcbinom(k, n, .481, log = TRUE)), dcbinom(k, n, .481))

[Package *cbinom* version 1.6 Index]