Logarithm of Cumulative Distribution Function

Rayleigh distribution logarithm of 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 logcdf = require( '@stdlib/stats/base/dists/rayleigh/logcdf' );

logcdf( x, sigma )

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

var y = logcdf( 2.0, 3.0 );
// returns ~-1.613

y = logcdf( 1.0, 2.0 );
// returns ~-2.141

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

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

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

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

If provided sigma < 0, the function returns NaN.

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

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

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

y = logcdf( 0.0, 0.0 );
// returns 0.0

y = logcdf( 2.0, 0.0 );
// returns 0.0

logcdf.factory( sigma )

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

var mylogCDF = logcdf.factory( 0.5 );
y = mylogCDF( 1.0 );
// returns ~-0.145

y = mylogCDF( 0.5 );
// returns ~-0.933

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 logcdf = require( '@stdlib/stats/base/dists/rayleigh/logcdf' );

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

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