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Probability Density Function

Truncated normal distribution probability density function (PDF).

A normally distributed random variable X conditional on a < X < b is called a truncated normal distribution. The probability density function (PDF) for a truncated normal random variable is

$$f(x;\mu,\sigma,a,b) = \begin{cases} \frac{\frac{1}{\sigma}\phi(\frac{x - \mu}{\sigma})}{\Phi(\frac{b - \mu}{\sigma}) - \Phi(\frac{a - \mu}{\sigma}) } & \text{ if } a < x < b \\ 0 & \text{ otherwise } \end{cases}$$

where Phi and phi denote the cumulative distribution function and density function of the normal distribution, respectively, mu is the location and sigma > 0 is the scale parameter of the distribution. a and b are the minimum and maximum support.

Installation

npm install @stdlib/stats-base-dists-truncated-normal-pdf

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

var pdf = require( '@stdlib/stats-base-dists-truncated-normal-pdf' );

pdf( x, a, b, mu, sigma )

Evaluates the probability density function (PDF) for a truncated normal distribution with lower limit a, upper limit b, location parameter mu, and scale parameter sigma.

var y = pdf( 0.9, 0.0, 1.0, 0.0, 1.0 );
// returns ~0.7795

y = pdf( 0.9, 0.0, 1.0, 0.5, 1.0 );
// returns ~0.9617

y = pdf( 0.9, -1.0, 1.0, 0.5, 1.0 );
// returns ~0.5896

y = pdf( 1.4, 0.0, 1.0, 0.0, 1.0 );
// returns 0.0

y = pdf( -0.9, 0.0, 1.0, 0.0, 1.0 );
// returns 0.0

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

var y = pdf( NaN, 0.0, 1.0, 0.5, 2.0 );
// returns NaN

y = pdf( 0.0, NaN, 1.0, 0.5, 2.0 );
// returns NaN

y = pdf( 0.0, 0.0, NaN, 0.5, 2.0 );
// returns NaN

y = pdf( 0.6, 0.0, 1.0, NaN, 2.0 );
// returns NaN

y = pdf( 0.6, 0.0, 1.0, 0.5, NaN );
// returns NaN

pdf.factory( a, b, mu, sigma )

Returns a function for evaluating the probability density function (PDF) for a truncated normal distribution.

var myPDF = pdf.factory( 0.0, 1.0, 0.0, 1.0 );
var y = myPDF( 0.8 );
// returns ~0.849

myPDF = pdf.factory( 0.0, 1.0, 0.5, 1.0 );
y = myPDF( 0.8 );
// returns ~0.996

Examples

var randu = require( '@stdlib/random-base-randu' );
var pdf = require( '@stdlib/stats-base-dists-truncated-normal-pdf' );

var sigma;
var mu;
var a;
var b;
var x;
var y;
var i;

for ( i = 0; i < 25; i++ ) {
    a = ( randu() * 80.0 ) - 40.0;
    b = a + ( randu() * 80.0 );
    x = ( randu() * 40.0 ) + a;
    mu = ( randu() * 20.0 ) - 10.0;
    sigma = ( randu() * 10.0 ) + 2.0;
    y = pdf( x, a, b, mu, sigma );
    console.log( 'x: %d, a: %d, b: %d, mu: %d, sigma: %d, f(x;a,b,mu,sigma): %d', x.toFixed( 4 ), a.toFixed( 4 ), b.toFixed( 4 ), mu.toFixed( 4 ), sigma.toFixed( 4 ), y.toFixed( 4 ) );
}

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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.

Community

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License

See LICENSE.

Copyright

Copyright © 2016-2024. The Stdlib Authors.