# @stdlib/stats-base-dists-negative-binomial-pmf

0.1.1 • Public • Published

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# Probability Mass Function

Negative binomial distribution probability mass function (PMF).

The probability mass function (PMF) for a negative binomial random variable `X` is

where `r > 0` is the number of successes until experiment is stopped and `0 < p <= 1` is the success probability. The random variable `X` denotes the number of failures until the `r` success is reached.

## Installation

`npm install @stdlib/stats-base-dists-negative-binomial-pmf`

## Usage

`var pmf = require( '@stdlib/stats-base-dists-negative-binomial-pmf' );`

#### pmf( x, r, p )

Evaluates the probability mass function for a negative binomial distribution with number of successes until experiment is stopped `r` and success probability `p`.

```var y = pmf( 5.0, 20.0, 0.8 );
// returns ~0.157

y = pmf( 21.0, 20.0, 0.5 );
// returns ~0.06

y = pmf( 5.0, 10.0, 0.4 );
// returns ~0.016

y = pmf( 0.0, 10.0, 0.9 );
// returns ~0.349```

While `r` can be interpreted as the number of successes until the experiment is stopped, the negative binomial distribution is also defined for non-integers `r`. In this case, `r` denotes shape parameter of the gamma mixing distribution.

```var y = pmf( 21.0, 15.5, 0.5 );
// returns ~0.037

y = pmf( 5.0, 7.4, 0.4 );
// returns ~0.051```

If provided a `r` which is not a positive number, the function returns `NaN`.

```var y = pmf( 2.0, 0.0, 0.5 );
// returns NaN

y = pmf( 2.0, -2.0, 0.5 );
// returns NaN```

If provided `NaN` as any argument, the function returns `NaN`.

```var y = pmf( NaN, 20.0, 0.5 );
// returns NaN

y = pmf( 0.0, NaN, 0.5 );
// returns NaN

y = pmf( 0.0, 20.0, NaN );
// returns NaN```

If provided a success probability `p` outside of `[0,1]`, the function returns `NaN`.

```var y = pmf( 2.0, 20, -1.0 );
// returns NaN

y = pmf( 2.0, 20, 1.5 );
// returns NaN```

#### pmf.factory( r, p )

Returns a function for evaluating the probability mass function (PMF) of a negative binomial distribution with number of successes until experiment is stopped `r` and success probability `p`.

```var mypmf = pmf.factory( 10, 0.5 );
var y = mypmf( 3.0 );
// returns ~0.03

y = mypmf( 10.0 );
// returns ~0.088```

## Examples

```var randu = require( '@stdlib/random-base-randu' );
var round = require( '@stdlib/math-base-special-round' );
var pmf = require( '@stdlib/stats-base-dists-negative-binomial-pmf' );

var i;
var r;
var p;
var x;
var y;

for ( i = 0; i < 10; i++ ) {
x = round( randu() * 30 );
r = randu() * 50;
p = randu();
y = pmf( x, r, p );
console.log( 'x: %d, r: %d, p: %d, P(X=x;r,p): %d', x, r, p.toFixed( 4 ), y.toFixed( 4 ) );
}```

## Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

## Package Sidebar

### Install

`npm i @stdlib/stats-base-dists-negative-binomial-pmf`

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