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Standard Deviation

Pareto (Type I) distribution standard deviation.

The standard deviation for a Pareto (Type I) random variable is

$$\mathop{\mathrm{SD}}\left( X \right) = \begin{cases} \infty & \text{for }\alpha\in(0,2] \\ \sqrt{ \frac{\beta^2\alpha}{(\alpha-1)^2(\alpha-2)} } & \text{for }\alpha>2 \end{cases}$$

where α > 0 is the shape parameter and β > 0 is the scale parameter.

Installation

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

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

stdev( alpha, beta )

Returns the standard deviation of a Pareto (Type I) distribution with parameters alpha (shape parameter) and beta (scale parameter).

var v = stdev( 2.5, 1.0 );
// returns ~1.491

v = stdev( 4.0, 12.0 );
// returns ~5.657

v = stdev( 8.0, 2.0 );
// returns ~0.33

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

var v = stdev( NaN, 2.0 );
// returns NaN

v = stdev( 2.0, NaN );
// returns NaN

If provided 0 < alpha <= 2, the function returns +Infinity.

var v = stdev( 0.5, 2.0 );
// returns Infinity

v = stdev( 1.5, 1.0 );
// returns Infinity

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

var v = stdev( 0.0, 1.0 );
// returns NaN

v = stdev( -1.0, 1.0 );
// returns NaN

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

var v = stdev( 1.0, 0.0 );
// returns NaN

v = stdev( 1.0, -1.0 );
// returns NaN

Examples

var randu = require( '@stdlib/random-base-randu' );
var EPS = require( '@stdlib/constants-float64-eps' );
var stdev = require( '@stdlib/stats-base-dists-pareto-type1-stdev' );

var alpha;
var beta;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    alpha = ( randu()*10.0 ) + EPS;
    beta = ( randu()*10.0 ) + EPS;
    v = stdev( alpha, beta );
    console.log( 'α: %d, β: %d, Var(X;α,β): %d', alpha.toFixed( 4 ), beta.toFixed( 4 ), v.toFixed( 4 ) );
}

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C APIs

Usage

#include "stdlib/stats/base/dists/pareto-type1/stdev.h"

stdlib_base_dists_pareto_type1_stdev( alpha, beta )

Returns the standard deviation of a Pareto (Type I) distribution.

double out = stdlib_base_dists_pareto_type1_stdev( 2.5, 1.0 );
// returns ~1.491

The function accepts the following arguments:

• alpha: [in] double first shape parameter.
• beta: [in] double second shape parameter.

double stdlib_base_dists_pareto_type1_stdev( const double alpha, const double beta );

Examples

#include "stdlib/stats/base/dists/pareto-type1/stdev.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 alpha;
    double beta;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        alpha = random_uniform( 0.0, 10.0 );
        beta = random_uniform( 0.0, 10.0 );
        y = stdlib_base_dists_pareto_type1_stdev( alpha, beta );
        printf( "α: %lf, β: %lf, SD(X;α,β): %lf\n", alpha, beta, y );
    }
}

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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.