feat: add Wald distribution variance package

PR-URL: #9730
Ref: #209

Reviewed-by: Philipp Burckhardt <[email protected]>
Reviewed-by: Neeraj Pathak <[email protected]>

Author: manit2004

lib/node_modules/@stdlib/stats/base/dists/wald/variance/README.md

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+<!--
+
+@license Apache-2.0
+
+Copyright (c) 2026 The Stdlib Authors.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+   http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+
+-->
+
+# Variance
+
+> [Wald][wald-distribution] distribution [variance][variance].
+
+<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->
+
+<section class="intro">
+
+The [variance][variance] for a [Wald][wald-distribution] random variable with mean `μ` and shape parameter `λ > 0` is
+
+<!-- <equation class="equation" label="eq:wald_variance" align="center" raw="\operatorname{Var}\left[ X \right] = \frac{\mu^{3}}{\lambda}" alt="Variance for a Wald distribution."> -->
+
+```math
+\operatorname{Var}\left[ X \right] = \frac{\mu^{3}}{\lambda}
+```
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
+<!-- Package usage documentation. -->
+
+<section class="usage">
+
+## Usage
+
+```javascript
+var variance = require( '@stdlib/stats/base/dists/wald/variance' );
+```
+
+#### variance( mu, lambda )
+
+Returns the [variance][variance] for a [Wald][wald-distribution] distribution with parameters `mu` (mean) and `lambda` (shape parameter).
+
+```javascript
+var y = variance( 2.0, 1.0 );
+// returns 8.0
+
+y = variance( 0.0, 1.0 );
+// returns NaN
+
+y = variance( 4.0, -1.0 );
+// returns NaN
+```
+
+If provided `NaN` as any argument, the function returns `NaN`.
+
+```javascript
+var y = variance( NaN, 1.0 );
+// returns NaN
+
+y = variance( 0.0, NaN );
+// returns NaN
+```
+
+If provided `mu <= 0` or `lambda <= 0`, the function returns `NaN`.
+
+```javascript
+var y = variance( 0.0, 0.0 );
+// returns NaN
+
+y = variance( 0.0, -1.0 );
+// returns NaN
+
+y = variance( -1.0, 0.0 );
+// returns NaN
+```
+
+</section>
+
+<!-- /.usage -->
+
+<!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
+
+<section class="notes">
+
+</section>
+
+<!-- /.notes -->
+
+<!-- Package usage examples. -->
+
+<section class="examples">
+
+## Examples
+
+<!-- eslint no-undef: "error" -->
+
+```javascript
+var uniform = require( '@stdlib/random/array/uniform' );
+var logEachMap = require( '@stdlib/console/log-each-map' );
+var EPS = require( '@stdlib/constants/float64/eps' );
+var variance = require( '@stdlib/stats/base/dists/wald/variance' );
+
+var opts = {
+    'dtype': 'float64'
+};
+var mu = uniform( 10, EPS, 10.0, opts );
+var lambda = uniform( 10, EPS, 20.0, opts );
+
+logEachMap( 'µ: %0.4f, λ: %0.4f, Var(X;µ,λ): %0.4f', mu, lambda, variance );
+```
+
+</section>
+
+<!-- /.examples -->
+
+<!-- C interface documentation. -->
+
+* * *
+
+<section class="c">
+
+## C APIs
+
+<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->
+
+<section class="intro">
+
+</section>
+
+<!-- /.intro -->
+
+<!-- C usage documentation. -->
+
+<section class="usage">
+
+### Usage
+
+```c
+#include "stdlib/stats/base/dists/wald/variance.h"
+```
+
+#### stdlib_base_dists_wald_variance( mu, lambda )
+
+Returns the [variance][variance] for a [Wald][wald-distribution] distribution with mean `mu` and shape parameter `lambda`.
+
+```c
+double out = stdlib_base_dists_wald_variance( 2.0, 1.0 );
+// returns 8.0
+```
+
+The function accepts the following arguments:
+
+-   **mu**: `[in] double` mean.
+-   **lambda**: `[in] double` shape parameter.
+
+```c
+double stdlib_base_dists_wald_variance( const double mu, const double lambda );
+```
+
+</section>
+
+<!-- /.usage -->
+
+<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
+
+<section class="notes">
+
+</section>
+
+<!-- /.notes -->
+
+<!-- C API usage examples. -->
+
+<section class="examples">
+
+### Examples
+
+```c
+#include "stdlib/stats/base/dists/wald/variance.h"
+#include "stdlib/constants/float64/eps.h"
+#include <stdlib.h>
+#include <stdio.h>
+
+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 lambda;
+    double mu;
+    double y;
+    int i;
+
+    for ( i = 0; i < 10; i++ ) {
+        mu = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 5.0 );
+        lambda = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 20.0 );
+        y = stdlib_base_dists_wald_variance( mu, lambda );
+        printf( "µ: %.4f, λ: %.4f, Var(X;µ,λ): %.4f\n", mu, lambda, y );
+    }
+}
+```
+
+</section>
+
+<!-- /.examples -->
+
+</section>
+
+<!-- /.c -->
+
+<!-- Section to include cited references. If references are included, add a horizontal rule *before* the section. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
+
+<section class="references">
+
+</section>
+
+<!-- /.references -->
+
+<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->
+
+<section class="related">
+
+</section>
+
+<!-- /.related -->
+
+<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
+
+<section class="links">
+
+[wald-distribution]: https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution
+
+[variance]: https://en.wikipedia.org/wiki/Variance
+
+</section>
+
+<!-- /.links -->
+

lib/node_modules/@stdlib/stats/base/dists/wald/variance/benchmark/benchmark.js

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+/**
+* @license Apache-2.0
+*
+* Copyright (c) 2026 The Stdlib Authors.
+*
+* Licensed under the Apache License, Version 2.0 (the "License");
+* you may not use this file except in compliance with the License.
+* You may obtain a copy of the License at
+*
+*    http://www.apache.org/licenses/LICENSE-2.0
+*
+* Unless required by applicable law or agreed to in writing, software
+* distributed under the License is distributed on an "AS IS" BASIS,
+* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+* See the License for the specific language governing permissions and
+* limitations under the License.
+*/
+
+'use strict';
+
+// MODULES //
+
+var bench = require( '@stdlib/bench' );
+var uniform = require( '@stdlib/random/array/uniform' );
+var isnan = require( '@stdlib/math/base/assert/is-nan' );
+var EPS = require( '@stdlib/constants/float64/eps' );
+var pkg = require( './../package.json' ).name;
+var variance = require( './../lib' );
+
+
+// MAIN //
+
+bench( pkg, function benchmark( b ) {
+	var lambda;
+	var len;
+	var mu;
+	var y;
+	var i;
+
+	len = 100;
+	mu = uniform( len, EPS, 100.0 );
+	lambda = uniform( len, EPS, 20.0 );
+
+	b.tic();
+	for ( i = 0; i < b.iterations; i++ ) {
+		y = variance( mu[ i%len ], lambda[ i%len ] );
+		if ( isnan( y ) ) {
+			b.fail( 'should not return NaN' );
+		}
+	}
+	b.toc();
+	if ( isnan( y ) ) {
+		b.fail( 'should not return NaN' );
+	}
+	b.pass( 'benchmark finished' );
+	b.end();
+});

lib/node_modules/@stdlib/stats/base/dists/wald/variance/benchmark/benchmark.native.js

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+/**
+* @license Apache-2.0
+*
+* Copyright (c) 2026 The Stdlib Authors.
+*
+* Licensed under the Apache License, Version 2.0 (the "License");
+* you may not use this file except in compliance with the License.
+* You may obtain a copy of the License at
+*
+*    http://www.apache.org/licenses/LICENSE-2.0
+*
+* Unless required by applicable law or agreed to in writing, software
+* distributed under the License is distributed on an "AS IS" BASIS,
+* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+* See the License for the specific language governing permissions and
+* limitations under the License.
+*/
+
+'use strict';
+
+// MODULES //
+
+var resolve = require( 'path' ).resolve;
+var bench = require( '@stdlib/bench' );
+var uniform = require( '@stdlib/random/array/uniform' );
+var isnan = require( '@stdlib/math/base/assert/is-nan' );
+var tryRequire = require( '@stdlib/utils/try-require' );
+var format = require( '@stdlib/string/format' );
+var EPS = require( '@stdlib/constants/float64/eps' );
+var pkg = require( './../package.json' ).name;
+
+
+// VARIABLES //
+
+var variance = tryRequire( resolve( __dirname, './../lib/native.js' ) );
+var opts = {
+	'skip': ( variance instanceof Error )
+};
+
+
+// MAIN //
+
+bench( format( '%s::native', pkg ), opts, function benchmark( b ) {
+	var lambda;
+	var len;
+	var mu;
+	var y;
+	var i;
+
+	len = 100;
+	mu = uniform( len, EPS, 100.0 );
+	lambda = uniform( len, EPS, 20.0 );
+
+	b.tic();
+	for ( i = 0; i < b.iterations; i++ ) {
+		y = variance( mu[ i%len ], lambda[ i%len ] );
+		if ( isnan( y ) ) {
+			b.fail( 'should not return NaN' );
+		}
+	}
+	b.toc();
+	if ( isnan( y ) ) {
+		b.fail( 'should not return NaN' );
+	}
+	b.pass( 'benchmark finished' );
+	b.end();
+});