Negative Binomial Distribution

Negative binomial distribution with parameters \(r\) (number of successful trials) and \(p\) (probability of success).

expValNBinom(
  size,
  prob = (1/(1 + beta)),
  beta = ((1 - prob)/prob),
  nb_tries = FALSE
)

varNBinom(
  size,
  prob = (1/(1 + beta)),
  beta = ((1 - prob)/prob),
  nb_tries = FALSE
)

mgfNBinom(
  t,
  size,
  prob = (1/(1 + beta)),
  beta = ((1 - prob)/prob),
  nb_tries = FALSE
)

pgfNBinom(
  t,
  size,
  prob = (1/(1 + beta)),
  beta = ((1 - prob)/prob),
  nb_tries = FALSE
)

Arguments

size	Number of successful trials.
prob	Probability of success in each trial.
beta	Alternative parameterization of the negative binomial distribution where beta = (1 - p) / p.
nb_tries	logical; if FALSE (default) number of trials until the rth success, otherwise, number of failures until the rth success.
t	t.

Value

Function :

• expValNBinom gives the expected value.

• varNBinom gives the variance.

• mgfNBinom gives the moment generating function (MGF).

• pgfNBinom gives the probability generating function (PGF).

Invalid parameter values will return an error detailing which parameter is problematic.

Details

When \(k\) is the number of failures until the \(r\)th success, with a probability \(p\) of a success, the negative binomial has density: $$\left(\frac{r + k - 1}{k}\right) (p)^{r} (1 - p)^{k}$$ for \(k \in \{0, 1, \dots \}\)

When \(k\) is the number of trials until the \(r\)th success, with a probability \(p\) of a success, the negative binomial has density: $$\left(\frac{k - 1}{r - 1}\right) (p)^{r} (1 - p)^{k - r}$$ for \(k \in \{r, r + 1, r + 2, \dots \}\)

The alternative parameterization of the negative binomial with parameter \(\beta\), and \(k\) being the number of trials, has density: $$\frac{\Gamma(r + k)}{\Gamma(r) k!} \left(\frac{1}{1 + \beta}\right)^{r}% \left(\frac{\beta}{1 + \beta}\right)^{k - r}$$ for \(k \in \{0, 1, \dots \}\)

Examples

# Where k is the number of trials for a rth success
expValNBinom(size = 2, prob = .4)
#> [1] 3

# Where k is the number of failures before a rth success
expValNBinom(size = 2, prob = .4, nb_tries = TRUE)
#> [1] 5

# With alternative parameterization where k is the number of trials
expValNBinom(size = 2, beta = 1.5)
#> [1] 3


# Where k is the number of trials for a rth success
varNBinom(size = 2, prob = .4)
#> [1] 7.5

# Where k is the number of failures before a rth success
varNBinom(size = 2, prob = .4, nb_tries = TRUE)
#> [1] 7.5

# With alternative parameterization where k is the number of trials
varNBinom(size = 2, beta = 1.5)
#> [1] 7.5

mgfNBinom(t = 1, size = 4, prob = 0.5)
#> [1] 3.756821

pgfNBinom(t = 5, size = 3, prob = 0.3)
#> [1] -0.001728