NBinom: Negative Binomial Distribution

In Distributacalcul: Probability Distribution Functions

Negative Binomial Distribution

Description

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

Usage

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.

Details

When k is the number of failures until the rth 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 rth 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 \}

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.

Examples

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

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

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


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

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

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

mgfNBinom(t = 1, size = 4, prob = 0.5)

pgfNBinom(t = 5, size = 3, prob = 0.3)

Distributacalcul documentation built on May 29, 2024, 9:25 a.m.