# NBinom: Negative Binomial Distribution In Distributacalcul: Probability Distribution Functions

## Description

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

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29``` ```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 `r`th success, otherwise, number of failures until the `r`th 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:

Pr(M = k) = choose(r + k - 1, k) p^r (1 - p)^k

for k = 0, 1, 2, ...

When k is the number of trials until the rth success, with a probability p of a success, the negative binomial has density:

Pr(M = k) = choose(k - 1, r - 1) p^r (1 - p)^(k - r)

for k = r, r + 1, r + 2, ...

The alternative parameterization of the negative binomial with parameter beta, and k being the number of trials, has density:

Pr(M = k) = (r + k - 1)!/((r - 1)! k!) (1/(1 + beta))^r (beta/(1 + beta))^(k - r)

for k = 0, 1, 2, ...

## 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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22``` ```# 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 Sept. 13, 2020, 5:19 p.m.