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Logarithm of Probability Density Function

Evaluate the natural logarithm of the probability density function (PDF) for a Pareto (Type I) distribution.

The probability density function (PDF) for a Pareto (Type I) random variable is

$$f(x;\alpha,\beta) = \begin{cases} \frac{\alpha\,\beta^\alpha}{x^{\alpha+1}} & \text{ for }x\ge \beta \\ 0 & \text{otherwise} \end{cases}$$

where alpha > 0 is the shape parameter and beta > 0 is the scale parameter.

Installation

npm install @stdlib/stats-base-dists-pareto-type1-logpdf

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 logpdf = require( '@stdlib/stats-base-dists-pareto-type1-logpdf' );

logpdf( x, alpha, beta )

Evaluates the natural logarithm of the probability density function (PDF) for a Pareto (Type I) distribution with parameters alpha (shape parameter) and beta (scale parameter).

var y = logpdf( 4.0, 1.0, 1.0 );
// returns ~-2.773

y = logpdf( 20.0, 1.0, 10.0 );
// returns ~-3.689

y = logpdf( 7.0, 2.0, 6.0 );
// returns ~-1.561

y = logpdf( 7.0, 6.0, 3.0 );
// returns ~-5.238

y = logpdf( 1.0, 4.0, 2.0 );
// returns -Infinity

y = logpdf( 1.5, 4.0, 2.0 );
// returns -Infinity

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

var y = logpdf( NaN, 1.0, 1.0 );
// returns NaN

y = logpdf( 0.0, NaN, 1.0 );
// returns NaN

y = logpdf( 0.0, 1.0, NaN );
// returns NaN

If provided alpha <= 0, the function returns NaN.

var y = logpdf( 2.0, -1.0, 0.5 );
// returns NaN

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

If provided beta <= 0, the function returns NaN.

var y = logpdf( 2.0, 0.5, -1.0 );
// returns NaN

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

logpdf.factory( alpha, beta )

Returns a function for evaluating the natural logarithm of the probability density function (PDF) (CDF) of a Pareto (Type I) distribution with parameters alpha (shape parameter) and beta (scale parameter).

var mylogpdf = logpdf.factory( 0.5, 0.5 );
var y = mylogpdf( 0.8 );
// returns ~-0.705

y = mylogpdf( 2.0 );
// returns ~-2.079

Notes

• In virtually all cases, using the logpdf or logcdf functions is preferable to manually computing the logarithm of the pdf or cdf, respectively, since the latter is prone to overflow and underflow.

Examples

var randu = require( '@stdlib/random-base-randu' );
var logpdf = require( '@stdlib/stats-base-dists-pareto-type1-logpdf' );

var alpha;
var beta;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 8.0;
    alpha = randu() * 4.0;
    beta = randu() * 4.0;
    y = logpdf( x, alpha, beta );
    console.log( 'x: %d, α: %d, β: %d, ln(f(x;α,β)): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), beta.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.