Negative Binomial Distribution Functions

Functions to compute the probability mass function (pmf), cumulative distribution function, and quantile function for the Negative Binomial distribution.

The PDF is:

$$f(k; r, p) = \frac{\Gamma(r + k)}{k!\Gamma(r)}p^r (1-p)^k$$

The CDF is:

$$F(k; r, p) = I_p(r, k + 1)$$

Where \(I_p\) is the regularised incomplete beta function.

Usage

negative_binomial_distribution(successes, success_fraction)

negative_binomial_pdf(x, successes, success_fraction)

negative_binomial_lpdf(x, successes, success_fraction)

negative_binomial_cdf(x, successes, success_fraction)

negative_binomial_lcdf(x, successes, success_fraction)

negative_binomial_quantile(p, successes, success_fraction)

negative_binomial_find_lower_bound_on_p(trials, successes, alpha)

negative_binomial_find_upper_bound_on_p(trials, successes, alpha)

negative_binomial_find_minimum_number_of_trials(
  failures,
  success_fraction,
  alpha
)

negative_binomial_find_maximum_number_of_trials(
  failures,
  success_fraction,
  alpha
)

Arguments

successes

Number of successes (successes > 0).

success_fraction

Probability of success on each trial (0 <= success_fraction <= 1).

x

Quantile value.

p

Probability (0 <= p <= 1).

trials

Number of trials.

alpha

Significance level (0 < alpha < 1).

failures

Number of failures (failures >= 0).

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See also

Boost Documentation for more details on the mathematical background.

Examples

# Negative Binomial distribution with successes = 5, success_fraction = 0.5
dist <- negative_binomial_distribution(5, 0.5)
# Apply generic functions
cdf(dist, 0.5)
#> [1] 0.06449911
logcdf(dist, 0.5)
#> [1] -2.741104
pdf(dist, 0.5)
#> [1] 0.05437955
logpdf(dist, 0.5)
#> [1] -2.911767
hazard(dist, 0.5)
#> [1] 0.05812881
chf(dist, 0.5)
#> [1] 0.06667318
mean(dist)
#> [1] 5
median(dist)
#> [1] 4
mode(dist)
#> [1] 4
range(dist)
#> [1]  0.000000e+00 1.797693e+308
quantile(dist, 0.2)
#> [1] 1
standard_deviation(dist)
#> [1] 3.162278
support(dist)
#> [1]  0.000000e+00 1.797693e+308
variance(dist)
#> [1] 10
skewness(dist)
#> [1] 0.9486833
kurtosis(dist)
#> [1] 4.3
kurtosis_excess(dist)
#> [1] 1.3

# Convenience functions
negative_binomial_pdf(3, 5, 0.5)
#> [1] 0.1367188
negative_binomial_lpdf(3, 5, 0.5)
#> [1] -1.989829
negative_binomial_cdf(3, 5, 0.5)
#> [1] 0.3632812
negative_binomial_lcdf(3, 5, 0.5)
#> [1] -1.012578
negative_binomial_quantile(0.5, 5, 0.5)
#> [1] 4

if (FALSE) { # \dontrun{
# Find lower bound on p given 10 trials and 5 successes with 95% confidence
negative_binomial_find_lower_bound_on_p(10, 5, 0.05)
# Find upper bound on p given 10 trials and 5 successes with 95% confidence
negative_binomial_find_upper_bound_on_p(10, 5, 0.05)
# Find minimum number of trials to observe 3 failures with success fraction 0.5 at 95% confidence
negative_binomial_find_minimum_number_of_trials(3, 0.5, 0.05)
# Find maximum number of trials to observe 3 failures with success fraction 0.5 at 95% confidence
negative_binomial_find_maximum_number_of_trials(3, 0.5, 0.05)
} # }