Cumulative Distribution Function

Rayleigh distribution cumulative distribution function.

The cumulative distribution function for a Rayleigh random variable is

upper F left-parenthesis x semicolon sigma right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column for x less-than 0 2nd Row 1st Column 1 minus e Superscript minus x squared slash 2 sigma squared Baseline 2nd Column for x greater-than-or-equal-to 0 EndLayout

where sigma > 0 is the scale parameter.

Usage

var cdf = require( '@stdlib/math/base/dists/rayleigh/cdf' );

cdf( x, sigma )

Evaluates the cumulative distribution function for a Rayleigh distribution with scale parameter sigma.

var y = cdf( 2.0, 3.0 );
// returns ~0.199

y = cdf( 1.0, 2.0 );
// returns ~0.118

y = cdf( -1.0, 4.0 );
// returns 0.0

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

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

y = cdf( 0.0, NaN );
// returns NaN

If provided sigma < 0, the function returns NaN.

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

If provided sigma = 0, the function evaluates the CDF of a degenerate distribution centered at 0.

var y = cdf( -2.0, 0.0 );
// returns 0.0

y = cdf( 0.0, 0.0 );
// returns 1.0

y = cdf( 2.0, 0.0 );
// returns 1.0

cdf.factory( sigma )

Returns a function for evaluating the cumulative distribution function of a Rayleigh distribution with parameter sigma (scale parameter).

var myCDF = cdf.factory( 0.5 );
y = myCDF( 1.0 );
// returns ~0.865

y = myCDF( 0.5 );
// returns ~0.393

Examples

var randu = require( '@stdlib/random/base/randu' );
var cdf = require( '@stdlib/math/base/dists/rayleigh/cdf' );

var sigma;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 10.0;
    sigma = randu() * 10.0;
    y = cdf( x, sigma );
    console.log( 'x: %d, σ: %d, F(x;σ): %d', x.toFixed( 4 ), sigma.toFixed( 4 ), y.toFixed( 4 ) );
}