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