fix: correct MGF in stats/base/dists/negative-binomial/mgf

lib/node_modules/@stdlib/stats/base/dists/negative-binomial/ctor/README.md

@@ -204,7 +204,7 @@ Evaluates the [moment-generating function][mgf] (MGF).
 var nbinomial = new NegativeBinomial( 4.0, 0.2 );
 
 var y = nbinomial.mgf( 0.1 );
-// returns ~1.66
+// returns ~8.878
 ```
 
 #### NegativeBinomial.prototype.pmf( x )

lib/node_modules/@stdlib/stats/base/dists/negative-binomial/ctor/docs/repl.txt

@@ -81,7 +81,7 @@
     > nbinomial.logpmf( 3.0 )
     ~-2.837
     > nbinomial.mgf( 0.2 )
-    ~36.675
+    ~7.405
     > nbinomial.pmf( 3.0 )
     ~0.059
     > nbinomial.quantile( 0.8 )

lib/node_modules/@stdlib/stats/base/dists/negative-binomial/ctor/lib/main.js

@@ -117,7 +117,7 @@
 * var y = nbinomial.cdf( 10.0 );
 * // returns ~0.013
 *
 * var v = nbinomial.mode;
 * // returns 36.0
 */
 function NegativeBinomial() {
@@ -182,7 +182,7 @@
 * @example
 * var nbinomial = new NegativeBinomial( 12.0, 0.4 );
 *
 * var v = nbinomial.kurtosis;
 * // returns ~0.522
 */
 setReadOnlyAccessor( NegativeBinomial.prototype, 'kurtosis', function get() {
@@ -200,7 +200,7 @@
 * @example
 * var nbinomial = new NegativeBinomial( 12.0, 0.4 );
 *
 * var v = nbinomial.mean;
 * // returns ~18.0
 */
 setReadOnlyAccessor( NegativeBinomial.prototype, 'mean', function get() {
@@ -218,7 +218,7 @@
 * @example
 * var nbinomial = new NegativeBinomial( 12.0, 0.4 );
 *
 * var v = nbinomial.mode;
 * // returns 16.0
 */
 setReadOnlyAccessor( NegativeBinomial.prototype, 'mode', function get() {
@@ -236,7 +236,7 @@
 * @example
 * var nbinomial = new NegativeBinomial( 12.0, 0.4 );
 *
 * var v = nbinomial.skewness;
 * // returns ~0.596
 */
 setReadOnlyAccessor( NegativeBinomial.prototype, 'skewness', function get() {
@@ -254,7 +254,7 @@
 * @example
 * var nbinomial = new NegativeBinomial( 12.0, 0.4 );
 *
 * var v = nbinomial.stdev;
 * // returns ~6.708
 */
 setReadOnlyAccessor( NegativeBinomial.prototype, 'stdev', function get() {
@@ -272,7 +272,7 @@
 * @example
 * var nbinomial = new NegativeBinomial( 12.0, 0.4 );
 *
 * var v = nbinomial.variance;
 * // returns ~45.0
 */
 setReadOnlyAccessor( NegativeBinomial.prototype, 'variance', function get() {
@@ -329,7 +329,7 @@
 * var nbinomial = new NegativeBinomial( 4.0, 0.2 );
 *
 * var v = nbinomial.mgf( 0.1 );
-* // returns ~1.66
+* // returns ~8.878
 */
 setReadOnly( NegativeBinomial.prototype, 'mgf', negativeBinomialMGF );
 

lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/README.md

@@ -28,20 +28,20 @@ limitations under the License.
 
 The [moment-generating function][mgf] for a [negative binomial][negative-binomial-distribution] random variable is
 
-<!-- <equation class="equation" label="eq:negative_binomial_mgf_function" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \biggl(\frac{\left( 1- p \right) e^t }{1 - p e^t}\biggr)^{\!r} \text{ for }t<-\log p" alt="Moment-generating function (MGF) for a negative binomial distribution."> -->
+<!-- <equation class="equation" label="eq:negative_binomial_mgf_function" align="center" raw="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \biggl(\frac{p}{1 - \left( 1-p \right) e^t}\biggr)^{\!r} \text{ for }t<-\log(1-p)" alt="Moment-generating function (MGF) for a negative binomial distribution."> -->
 
 ```math
-M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \biggl(\frac{\left( 1- p \right) e^t }{1 - p e^t}\biggr)^{\!r} \text{ for }t<-\log p
+M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \biggl(\frac{p}{1 - \left( 1-p \right) e^t}\biggr)^{\!r} \text{ for }t<-\log(1-p)
 ```
 
-<!-- <div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \biggl(\frac{\left( 1- p \right) e^t }{1 - p e^t}\biggr)^{\!r} \text{ for }t&lt;-\log p" data-equation="eq:negative_binomial_mgf_function">
+<!-- <div class="equation" align="center" data-raw-text="M_X(t) := \mathbb{E}\!\left[e^{tX}\right] =  \biggl(\frac{p}{1 - \left( 1-p \right) e^t}\biggr)^{\!r} \text{ for }t&lt;-\log(1-p)" data-equation="eq:negative_binomial_mgf_function">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/docs/img/equation_negative_binomial_mgf_function.svg" alt="Moment-generating function (MGF) for a negative binomial distribution.">
     <br>
 </div> -->
 
 <!-- </equation> -->
 
-where `r > 0` is the number of failures until the experiment is stopped and `0 <= p <= 1` is the success probability.
+where `r > 0` is the number of successes until the experiment is stopped and `0 <= p <= 1` is the success probability.
 
 </section>
 
@@ -63,26 +63,26 @@ Evaluates the [moment-generating function][mgf] for a [negative binomial][negati
 
 ```javascript
 var y = mgf( 0.05, 20.0, 0.8 );
-// returns ~267.839
+// returns ~1.294
 
-y = mgf( 0.1, 20.0, 0.1 );
-// returns ~9.347
+y = mgf( 0.1, 20.0, 0.9 );
+// returns ~1.265
 ```
 
 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
+// returns ~5.598
 
-y = mgf( 0.5, 7.4, 0.4 );
-// returns ~2675.677
+y = mgf( 0.5, 7.4, 0.8 );
+// returns ~3.704
 ```
 
-If `t >= -ln( p )`, the function returns `NaN`.
+If `t >= -ln( 1-p )`, the function returns `NaN`.
 
 ```javascript
-var y = mgf( 0.7, 15.5, 0.5 ); // -ln( p ) = ~0.693
+var y = mgf( 0.7, 15.5, 0.5 ); // -ln( 1-p ) = ~0.693
 // returns NaN
 ```
 
@@ -126,10 +126,10 @@ Returns a function for evaluating the [moment-generating function][mgf] of a [ne
 ```javascript
 var myMGF = mgf.factory( 4.3, 0.4 );
 var y = myMGF( 0.2 );
-// returns ~4.696
+// returns ~5.672
 
 y = myMGF( 0.4 );
-// returns ~30.83
+// returns ~315.814
 ```
 
 </section>
@@ -203,7 +203,7 @@ Evaluates the [moment-generating function][mgf] for a [negative binomial][negati
 
 ```c
 double out = stdlib_base_dists_negative_binomial_mgf( 0.05, 20.0, 0.8 );
-// returns ~267.839
+// returns ~1.294
 ```
 
 The function accepts the following arguments:

lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/benchmark/benchmark.js

@@ -48,7 +48,7 @@ bench( pkg, function benchmark( b ) {
 	p = new Float64Array( len );
 	for ( i = 0; i < len; i++ ) {
 		p[ i ] = uniform( EPS, 1.0 );
-		t[ i ] = uniform( 0.0, -ln( p[ i ] ) );
+		t[ i ] = uniform( 0.0, -ln( 1.0 - p[ i ] ) );
 		r[ i ] = discreteUniform( 1, 100 );
 	}
 
@@ -82,7 +82,7 @@ bench( format( '%s::factory', pkg ), function benchmark( b ) {
 	len = 100;
 	t = new Float64Array( len );
 	for ( i = 0; i < len; i++ ) {
-		t[ i ] = uniform( 0.0, -ln( p ) );
+		t[ i ] = uniform( 0.0, -ln( 1.0 - p ) );
 	}
 
 	b.tic();

lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/benchmark/benchmark.native.js

@@ -57,7 +57,7 @@ bench( format( '%s::native', pkg ), opts, function benchmark( b ) {
 	p = new Float64Array( len );
 	for ( i = 0; i < len; i++ ) {
 		p[ i ] = uniform( EPS, 1.0 );
-		t[ i ] = uniform( 0.0, -ln( p[ i ] ) );
+		t[ i ] = uniform( 0.0, -ln( 1.0 - p[ i ] ) );
 		r[ i ] = discreteUniform( 1, 100 );
 	}
 

lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/docs/repl.txt

@@ -31,11 +31,11 @@
     Examples
     --------
     > var y = {{alias}}( 0.05, 20.0, 0.8 )
-    ~267.839
-    > y = {{alias}}( 0.1, 20.0, 0.1 )
-    ~9.347
-    > y = {{alias}}( 0.5, 10.0, 0.4 )
-    ~42822.023
+    ~1.294
+    > y = {{alias}}( 0.1, 20.0, 0.9 )
+    ~1.265
+    > y = {{alias}}( 0.5, 10.0, 0.8 )
+    ~5.868
 
     > y = {{alias}}( 0.1, 0.0, 0.5 )
     NaN
@@ -77,9 +77,9 @@
     --------
     > var myMGF = {{alias}}.factory( 4.3, 0.4 );
     > var y = myMGF( 0.2 )
-    ~4.696
+    ~5.672
     > y = myMGF( 0.4 )
-    ~30.83
+    ~315.814
 
     See Also
     --------

lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/docs/types/index.d.ts

@@ -45,15 +45,15 @@ interface MGF {
 	*
 	* @example
 	* var y = mgf( 0.05, 20.0, 0.8 );
-	* // returns ~267.839
+	* // returns ~1.294
 	*
 	* @example
-	* var y = mgf( 0.1, 20.0, 0.1 );
-	* // returns ~9.347
+	* var y = mgf( 0.1, 20.0, 0.9 );
+	* // returns ~1.265
 	*
 	* @example
-	* var y = mgf( 0.5, 10.0, 0.4 );
-	* // returns ~42822.023
+	* var y = mgf( 0.5, 10.0, 0.8 );
+	* // returns ~5.868
 	*
 	* @example
 	* var y = mgf( 0.1, 0.0, 0.5 );
@@ -95,7 +95,7 @@ interface MGF {
 	* @example
 	* var myMGF = mgf.factory( 4.3, 0.4 );
 	* var y = myMGF( 0.2 );
-	* // returns ~4.696
+	* // returns ~5.672
 	*/
 	factory( r: number, p: number ): Unary;
 }
@@ -110,20 +110,20 @@ interface MGF {
 *
 * @example
 * var y = mgf( 0.05, 20.0, 0.8 );
-* // returns ~267.839
+* // returns ~1.294
 *
-* y = mgf( 0.1, 20.0, 0.1 );
-* // returns ~9.347
+* y = mgf( 0.1, 20.0, 0.9 );
+* // returns ~1.265
 *
-* y = mgf( 0.5, 10.0, 0.4 );
-* // returns ~42822.023
+* y = mgf( 0.5, 10.0, 0.8 );
+* // returns ~5.868
 *
 * var myMGF = mgf.factory( 4.3, 0.4 );
 * y = myMGF( 0.2 );
-* // returns ~4.696
+* // returns ~5.672
 *
 * y = myMGF( 0.4 );
-* // returns ~30.83
+* // returns ~315.814
 */
 declare var mgf: MGF;
 

lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/factory.js

@@ -22,9 +22,9 @@
 
 var constantFunction = require( '@stdlib/utils/constant-function' );
 var isnan = require( '@stdlib/math/base/assert/is-nan' );
-var exp = require( '@stdlib/math/base/special/exp' );
+var expm1 = require( '@stdlib/math/base/special/expm1' );
 var pow = require( '@stdlib/math/base/special/pow' );
-var ln = require( '@stdlib/math/base/special/ln' );
+var log1p = require( '@stdlib/math/base/special/log1p' );
 
 
 // MAIN //
@@ -39,7 +39,7 @@ var ln = require( '@stdlib/math/base/special/ln' );
 * @example
 * var mgf = factory( 4.3, 0.4 );
 * var y = mgf( 0.2 );
-* // returns ~4.696
+* // returns ~5.672
 */
 function factory( r, p ) {
 	if (
@@ -65,10 +65,10 @@ function factory( r, p ) {
 	* // returns <number>
 	*/
 	function mgf( t ) {
-		if ( isnan( t ) || t >= -ln( p ) ) {
+		if ( isnan( t ) || t >= -log1p( -p ) ) {
 			return NaN;
 		}
-		return pow( ( (1.0 - p) * exp( t ) ) / ( 1.0 - (p * exp( t )) ), r );
+		return pow( p / ( p - ((1.0 - p) * expm1( t )) ), r );
 	}
 }
 

lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/index.js

@@ -27,20 +27,20 @@
 * var mgf = require( '@stdlib/stats/base/dists/negative-binomial/mgf' );
 *
 * var y = mgf( 0.05, 20.0, 0.8 );
-* // returns ~267.839
+* // returns ~1.294
 *
-* y = mgf( 0.1, 20.0, 0.1 );
-* // returns ~9.347
+* y = mgf( 0.1, 20.0, 0.9 );
+* // returns ~1.265
 *
-* y = mgf( 0.5, 10.0, 0.4 );
-* // returns ~42822.023
+* y = mgf( 0.5, 10.0, 0.8 );
+* // returns ~5.868
 *
 * var myMGF = mgf.factory( 4.3, 0.4 );
 * y = myMGF( 0.2 );
-* // returns ~4.696
+* // returns ~5.672
 *
 * y = myMGF( 0.4 );
-* // returns ~30.83
+* // returns ~315.814
 */
 
 // MODULES //

lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/main.js

@@ -21,9 +21,9 @@
 // MODULES //
 
 var isnan = require( '@stdlib/math/base/assert/is-nan' );
-var exp = require( '@stdlib/math/base/special/exp' );
+var expm1 = require( '@stdlib/math/base/special/expm1' );
 var pow = require( '@stdlib/math/base/special/pow' );
-var ln = require( '@stdlib/math/base/special/ln' );
+var log1p = require( '@stdlib/math/base/special/log1p' );
 
 
 // MAIN //
@@ -38,15 +38,15 @@ var ln = require( '@stdlib/math/base/special/ln' );
 *
 * @example
 * var y = mgf( 0.05, 20.0, 0.8 );
-* // returns ~267.839
+* // returns ~1.294
 *
 * @example
-* var y = mgf( 0.1, 20.0, 0.1 );
-* // returns ~9.347
+* var y = mgf( 0.1, 20.0, 0.9 );
+* // returns ~1.265
 *
 * @example
-* var y = mgf( 0.5, 10.0, 0.4 );
-* // returns ~42822.023
+* var y = mgf( 0.5, 10.0, 0.8 );
+* // returns ~5.868
 *
 * @example
 * var y = mgf( 0.1, 0.0, 0.5 );
@@ -84,11 +84,11 @@ function mgf( t, r, p ) {
 		r <= 0.0 ||
 		p < 0.0 ||
 		p > 1.0 ||
-		t >= -ln( p )
+		t >= -log1p( -p )
 	) {
 		return NaN;
 	}
-	return pow( ( (1.0 - p) * exp( t ) ) / ( 1.0 - (p * exp( t )) ), r );
+	return pow( p / ( p - ((1.0 - p) * expm1( t )) ), r );
 }
 
 

lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/lib/native.js

@@ -36,15 +36,15 @@ var addon = require( './../src/addon.node' );
 *
 * @example
 * var y = mgf( 0.05, 20.0, 0.8 );
-* // returns ~267.839
+* // returns ~1.294
 *
 * @example
-* var y = mgf( 0.1, 20.0, 0.1 );
-* // returns ~9.347
+* var y = mgf( 0.1, 20.0, 0.9 );
+* // returns ~1.265
 *
 * @example
-* var y = mgf( 0.5, 10.0, 0.4 );
-* // returns ~42822.023
+* var y = mgf( 0.5, 10.0, 0.8 );
+* // returns ~5.868
 *
 * @example
 * var y = mgf( 0.1, 0.0, 0.5 );

lib/node_modules/@stdlib/stats/base/dists/negative-binomial/mgf/manifest.json

@@ -40,9 +40,9 @@
       "dependencies": [
         "@stdlib/math/base/napi/ternary",
         "@stdlib/math/base/assert/is-nan",
-        "@stdlib/math/base/special/exp",
+        "@stdlib/math/base/special/expm1",
         "@stdlib/math/base/special/pow",
-        "@stdlib/math/base/special/ln"
+        "@stdlib/math/base/special/log1p"
       ]
     },
     {
@@ -58,9 +58,9 @@
       "libpath": [],
       "dependencies": [
         "@stdlib/math/base/assert/is-nan",
-        "@stdlib/math/base/special/exp",
+        "@stdlib/math/base/special/expm1",
         "@stdlib/math/base/special/pow",
-        "@stdlib/math/base/special/ln"
+        "@stdlib/math/base/special/log1p"
       ]
     },
     {
@@ -76,9 +76,9 @@
       "libpath": [],
       "dependencies": [
         "@stdlib/math/base/assert/is-nan",
-        "@stdlib/math/base/special/exp",
+        "@stdlib/math/base/special/expm1",
         "@stdlib/math/base/special/pow",
-        "@stdlib/math/base/special/ln"
+        "@stdlib/math/base/special/log1p"
       ]
     }
   ]