

   TThhee NNeeggaattiivvee BBiinnoommiiaall DDiissttrriibbuuttiioonn

        dnbinom(x, size, prob)
        pnbinom(q, size, prob)
        qnbinom(p, size, prob)
        rnbinom(n, size, prob)

   AArrgguummeennttss::

        x,q: vector of quantiles representing the number of
             failures which occur in a sequence of Bernoulli
             trials before a target number of successes is
             reached, or alternately the probability distribu-
             tion of a heterogeneous Poisson process whose
             intensity is distributed as a gamma distribution
             with scale parameter `prob/(1-prob)' and shape
             parameter `size' (this definition allows non-inte-
             ger values of `size').

          x: vector of (non-negative integer) quantiles.

          q: vector of quantiles.

          p: vector of probabilities.

          n: number of observations to generate.

       size: target for number of successful trials /
             shape parameter of gamma distribution.

       prob: probability of success in each trial /
             determines scale of gamma distribution (`prob' =
             `scale/(1+scale)').

   DDeessccrriippttiioonn::

        These functions provide information about the negative
        binomial distribution with parameters `size' and
        `prob'.  `dnbinom' gives the density, `pnbinom' gives
        the distribution function, `qnbinom' gives the quantile
        function and `rnbinom' generates random deviates.

   DDeettaaiillss::

        The negative binomial distribution with `size' = n and
        `prob' = p has density

              p(x) = Gamma(x+n)/(Gamma(n) x!) p^n (1-p)^x

        for x = 0, 1, 2, ...

        If an element of `x' is not integer, the result of
        `dnbinom' is zero, with a warning.

   SSeeee AAllssoo::

        `dbinom' for the binomial, `dpois' for the Poisson and
        `dgeom' for the geometric distribution, which is a spe-
        cial case of the negative binomial.

   EExxaammpplleess::

        x <- 0:11
        dnbinom(x, size = 1, prob = 1/2) * 2^(1 + x) # == 1
        126 /  dnbinom(0:8, size  = 2, prob  = 1/2) #- theoretically integer

        ## Cumulative ('p') = Sum of discrete prob.s ('d');  Relative error :
        summary(1 - cumsum(dnbinom(x, size = 2, prob = 1/2)) /
                       pnbinom(x, size  = 2, prob = 1/2))

        x <- 0:15
        size <- (1:20)/4
        persp(x,size, dnb <- outer(x,size,function(x,s)dnbinom(x,s, pr= 0.4)))
        title(tit <- "negative binomial density(x,s, pr = 0.4)  vs.  x & s")
        ## if persp() only could label axes ....

        image  (x,size, log10(dnb), main= paste("log [",tit,"]"))
        contour(x,size, log10(dnb),add=TRUE)

