## Usage ```javascript var mgf = require( '@stdlib/stats-base-dists-negative-binomial-mgf' ); ``` #### mgf( t, r, p ) Evaluates the [moment-generating function][mgf] for a [negative binomial][negative-binomial-distribution] distribution with number of successes until experiment is stopped `r` and success probability `p`. ```javascript var y = mgf( 0.05, 20.0, 0.8 ); // returns ~267.839 y = mgf( 0.1, 20.0, 0.1 ); // returns ~9.347 ``` While `r` can be interpreted as the number of successes until the experiment is stopped, the [negative binomial][negative-binomial-distribution] distribution is also defined for non-integers `r`. In this case, `r` denotes shape parameter of the [gamma mixing distribution][negative-binomial-mixture-representation]. ```javascript var y = mgf( 0.1, 15.5, 0.5 ); // returns ~26.375 y = mgf( 0.5, 7.4, 0.4 ); // returns ~2675.677 ``` If `t >= -ln( p )`, the function returns `NaN`. ```javascript var y = mgf( 0.7, 15.5, 0.5 ); // -ln( p ) = ~0.693 // returns NaN ``` If provided a `r` which is not a positive number, the function returns `NaN`. ```javascript var y = mgf( 0.2, 0.0, 0.5 ); // returns NaN y = mgf( 0.2, -2.0, 0.5 ); // returns NaN ``` If provided `NaN` as any argument, the function returns `NaN`. ```javascript var y = mgf( NaN, 20.0, 0.5 ); // returns NaN y = mgf( 0.0, NaN, 0.5 ); // returns NaN y = mgf( 0.0, 20.0, NaN ); // returns NaN ``` If provided a success probability `p` outside of `[0,1]`, the function returns `NaN`. ```javascript var y = mgf( 0.2, 20, -1.0 ); // returns NaN y = mgf( 0.2, 20, 1.5 ); // returns NaN ``` #### mgf.factory( r, p ) Returns a function for evaluating the [moment-generating function][mgf] of a [negative binomial][negative-binomial-distribution] distribution with number of successes until experiment is stopped `r` and success probability `p`. ```javascript var myMGF = mgf.factory( 4.3, 0.4 ); var y = myMGF( 0.2 ); // returns ~4.696 y = myMGF( 0.4 ); // returns ~30.83 ```

## Examples ```javascript var uniform = require( '@stdlib/random-array-uniform' ); var logEachMap = require( '@stdlib/console-log-each-map' ); var mgf = require( '@stdlib/stats-base-dists-negative-binomial-mgf' ); var opts = { 'dtype': 'float64' }; var t = uniform( 10, -0.5, 0.5, opts ); var r = uniform( 10, 0.0, 50.0, opts ); var p = uniform( 10, 0.0, 1.0, opts ); logEachMap( 't: %0.4f, r: %0.4f, p: %0.4f, M_X(t;r,p): %0.4f', t, r, p, mgf ); ```

## C APIs

### Usage ```c #include "stdlib/stats/base/dists/negative-binomial/mgf.h" ``` #### stdlib_base_dists_negative_binomial_mgf( t, r, p ) Evaluates the [moment-generating function][mgf] for a [negative binomial][negative-binomial-distribution] distribution with number of successes until experiment is stopped `r` and success probability `p`. ```c double out = stdlib_base_dists_negative_binomial_mgf( 0.05, 20.0, 0.8 ); // returns ~267.839 ``` The function accepts the following arguments: - **t**: `[in] double` input value. - **r**: `[in] double` number of successes until experiment is stopped. - **p**: `[in] double` success probability. ```c double stdlib_base_dists_negative_binomial_mgf( const double t, const double r, const double p ); ```

### Examples ```c #include "stdlib/stats/base/dists/negative-binomial/mgf.h" #include #include static double random_uniform( const double min, const double max ) { double v = (double)rand() / ( (double)RAND_MAX + 1.0 ); return min + ( v * ( max - min ) ); } int main( void ) { double t; double r; double p; double y; int i; for ( i = 0; i < 25; i++ ) { t = random_uniform( -1.0, 1.0 ); r = random_uniform( 1.0, 10.0 ); p = random_uniform( 0.0, 1.0 ); y = stdlib_base_dists_negative_binomial_mgf( t, r, p ); printf( "t: %lf, r: %lf, p: %lf, M_X(t;r,p): %lf\n", t, r, p, y ); } } ```

* * * ## Notice This package is part of [stdlib][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][stdlib], see the main project [repository][stdlib]. #### Community [![Chat][chat-image]][chat-url] --- ## License See [LICENSE][stdlib-license]. ## Copyright Copyright © 2016-2026. The Stdlib [Authors][stdlib-authors].

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