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* @license Apache-2.0
*
* Copyright (c) 2025 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 exp = require( '@stdlib/math/base/special/exp' );
var pow = require( '@stdlib/math/base/special/pow' );
var sqrt = require( '@stdlib/math/base/special/sqrt' );
var TWO_PI = require( '@stdlib/constants/float64/two-pi' );
var PINF = require( '@stdlib/constants/float64/pinf' );
var isnan = require( '@stdlib/math/base/assert/is-nan' );
// MAIN //
/**
* Evaluates the probability density function (PDF) for a Wald distribution with mean `mu` and shape parameter `lambda` at a value `x`.
*
* @param {number} x - input value
* @param {PositiveNumber} mu - mean
* @param {PositiveNumber} lambda - shape parameter
* @returns {number} evaluated probability density function
*
* @example
* var y = pdf( 2.0, 1.0, 1.0 );
* // returns ~ 0.110
*
* @example
* var y = pdf( 0.5, 2.0, 3.0 );
* // returns ~ 0.362
*
* @example
* var y = pdf( NaN, 1.0, 1.0 );
* // returns NaN
*
* @example
* var y = pdf( 1.0, NaN, 1.0 );
* // returns NaN
*
* @example
* var y = pdf( 1.0, 1.0, NaN );
* // returns NaN
*
* @example
* // Non-positive mean:
* var y = pdf( 2.0, 0.0, 1.0 );
* // returns NaN
*
* @example
* var y = pdf( 2.0, -1.0, 1.0 );
* // returns NaN
*
* @example
* // Negative shape parameter:
* var y = pdf( 2.0, 1.0, -1.0 );
* // returns NaN
*
* @example
* // Zero shape parameter (degenerate distribution):
* var y = pdf( 2.0, 1.0, 0.0 );
* // returns 0.0
*
* @example
* var y = pdf( 1.0, 1.0, 0.0 );
* // returns Infinity
*
* @example
* // Non-positive x:
* var y = pdf( 0.0, 1.0, 1.0 );
* // returns 0.0
*
* @example
* var y = pdf( -1.0, 1.0, 1.0 );
* // returns 0.0
*/
function pdf( x, mu, lambda ) {
var A;
var B;
if (
isnan( x ) ||
isnan( mu ) ||
isnan( lambda ) ||
mu <= 0.0 ||
lambda < 0.0
) {
return NaN;
}
if ( lambda === 0.0 ) {
return ( x === mu ) ? PINF : 0.0;
}
if ( x <= 0.0 ) {
return 0.0;
}
if ( !isFinite( x ) ) {
return 0.0;
}
A = sqrt( lambda / TWO_PI );
B = -lambda / ( 2.0 * pow( mu, 2.0 ) );
return A / sqrt( pow( x, 3.0 ) ) * exp( B * pow( x - mu, 2.0 ) / x );
}
// EXPORTS //
module.exports = pdf;
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