Let . Since
(1)
integration from zero to gives . Letting and imposing unit total probability gives
(2)
The probability density function is positive for because . Consequently the cumulative distribution function is increasing, starts at zero and tends to one. This verifies that the derived normalizing constant defines a probability distribution. The family is the exponential-logarithmic distribution with the complementary mixing parameter .
Solving gives . Hence the quantile function is
(1)
For such , , making the logarithm's argument lie in and the output positive. Its endpoint limits are and . Substitution into (i)'s cumulative distribution function recovers .
The following R function performs inverse transform sampling with vectorized numerical computation. It checks finite real scalar parameters, including a nonnegative integer sample count, and gives beta its requested default.
r_sample_f <- function(n, p, beta = 1) {
  scalar <- function(v) {
    is.numeric(v) && !is.complex(v) && length(v) == 1L &&
      is.finite(v)
  }
  if (!scalar(n) || n < 0 || n != floor(n))
    stop("n must be a nonnegative integer")
  if (!scalar(p) || p <= 0 || p >= 1)
    stop("p must lie strictly between 0 and 1")
  if (!scalar(beta) || beta <= 0)
    stop("beta must be positive")
  if (n == 0) return(numeric(0))
  v <- runif(n)
  (log1p(-p) - log(-expm1(v * log(p)))) / beta
}
Here is again a uniform distribution draw, so the expression is the quantile function from (ii). The function expm1(t) evaluates accurately when is small, avoiding cancellation in ; log1p(-p) evaluates . For default bounds, runif excludes both endpoints, so this implementation does not deliberately generate the infinite endpoint of the quantile function. It returns a numeric vector of length n, including an empty vector when n = 0. [ref] [ref]

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