# Nbinom-class

##### Class "Nbinom"

The negative binomial distribution with `size`

\(= n\), by default \(=1\), and
`prob`

\(= p\), by default \(=0.5\), has density
$$
d(x) = \frac{\Gamma(x+n)}{\Gamma(n) x!} p^n (1-p)^x$$
for \(x = 0, 1, 2, \ldots\)

This represents the number of failures
which occur in a sequence of Bernoulli trials before a target number
of successes is reached.
C.f. `rnbinom`

- Keywords
- distribution

##### Note

Working with a computer, we use a finite interval as support which carries at least mass `1-getdistrOption("TruncQuantile")`

.

##### Objects from the Class

Objects can be created by calls of the form `Nbinom(prob, size)`

.
This object is a negative binomial distribution.

##### Slots

`img`

Object of class

`"Naturals"`

: The space of the image of this distribution has got dimension 1 and the name "Natural Space".`param`

Object of class

`"NbinomParameter"`

: the parameter of this distribution (prob, size), declared at its instantiation`r`

Object of class

`"function"`

: generates random numbers (calls function`rnbinom`

)`d`

Object of class

`"function"`

: density function (calls function`dnbinom`

)`p`

Object of class

`"function"`

: cumulative function (calls function`pnbinom`

)`q`

Object of class

`"function"`

: inverse of the cumulative function (calls function`qnbinom`

). The quantile is defined as the smallest value \(x\) such that \(F(x) \ge p\), where \(F\) is the distribution function.`support`

Object of class

`"numeric"`

: a (sorted) vector containing the support of the discrete density function`.withArith`

logical: used internally to issue warnings as to interpretation of arithmetics

`.withSim`

logical: used internally to issue warnings as to accuracy

`.logExact`

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

`.lowerExact`

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

`Symmetry`

object of class

`"DistributionSymmetry"`

; used internally to avoid unnecessary calculations.

##### Extends

Class `"DiscreteDistribution"`

, directly.
Class `"UnivariateDistribution"`

, by class `"DiscreteDistribution"`

.
Class `"Distribution"`

, by class `"DiscreteDistribution"`

.

##### Methods

- initialize
`signature(.Object = "Nbinom")`

: initialize method- prob
`signature(object = "Nbinom")`

: returns the slot`prob`

of the parameter of the distribution- prob<-
`signature(object = "Nbinom")`

: modifies the slot`prob`

of the parameter of the distribution- size
`signature(object = "Nbinom")`

: returns the slot`size`

of the parameter of the distribution- size<-
`signature(object = "Nbinom")`

: modifies the slot`size`

of the parameter of the distribution- +
`signature(e1 = "Nbinom", e2 = "Nbinom")`

: For the negative binomial distribution we use its closedness under convolutions.

##### See Also

`NbinomParameter-class`

`Geom-class`

`DiscreteDistribution-class`

`Naturals-class`

`rnbinom`

##### Examples

```
# NOT RUN {
N <- Nbinom(prob = 0.5, size = 1) # N is a binomial distribution with prob=0.5 and size=1.
r(N)(1) # one random number generated from this distribution, e.g. 3
d(N)(1) # Density of this distribution is 0.25 for x=1.
p(N)(0.4) # Probability that x<0.4 is 0.5.
q(N)(.1) # x=0 is the smallest value x such that p(B)(x)>=0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
size(N) # size of this distribution is 1.
size(N) <- 2 # size of this distribution is now 2.
# }
```

*Documentation reproduced from package distr, version 2.8.0, License: LGPL-3*