Logarithm of Cumulative Distribution Function
Cauchy distribution logarithm of cumulative distribution function.
The cumulative distribution function for a Cauchy random variable is
where x0
is the location parameter and gamma > 0
is the scale parameter.
Usage
var logcdf = require( '@stdlib/stats/base/dists/cauchy/logcdf' );
logcdf( x, x0, gamma )
Evaluates the natural logarithm of the cumulative distribution function (CDF) for a Cauchy distribution with parameters x0
(location parameter) and gamma > 0
(scale parameter).
var y = logcdf( 4.0, 0.0, 2.0 );
// returns ~-0.16
y = logcdf( 1.0, 0.0, 2.0 );
// returns ~-0.435
y = logcdf( 1.0, 3.0, 2.0 );
// returns ~-1.386
If provided NaN
as any argument, the function returns NaN
.
var y = logcdf( NaN, 0.0, 2.0 );
// returns NaN
y = logcdf( 1.0, 2.0, NaN );
// returns NaN
y = logcdf( 1.0, NaN, 3.0 );
// returns NaN
If provided gamma <= 0
, the function returns NaN
.
var y = logcdf( 2.0, 0.0, -1.0 );
// returns NaN
y = logcdf( 2.0, 0.0, 0.0 );
// returns NaN
logcdf.factory( x0, gamma )
Returns a function for evaluating the natural logarithm of the cumulative distribution function of a Cauchy distribution with parameters x0
(location parameter) and gamma > 0
(scale parameter).
var mylogcdf = logcdf.factory( 10.0, 2.0 );
var y = mylogcdf( 10.0 );
// returns ~-0.693
y = mylogcdf( 12.0 );
// returns ~-0.288
Notes
- In virtually all cases, using the
logpdf
orlogcdf
functions is preferable to manually computing the logarithm of thepdf
orcdf
, respectively, since the latter is prone to overflow and underflow.
Examples
var randu = require( '@stdlib/random/base/randu' );
var EPS = require( '@stdlib/constants/float64/eps' );
var logcdf = require( '@stdlib/stats/base/dists/cauchy/logcdf' );
var gamma;
var x0;
var x;
var y;
var i;
for ( i = 0; i < 10; i++ ) {
x = randu() * 10.0;
x0 = randu() * 10.0;
gamma = ( randu()*10.0 ) + EPS;
y = logcdf( x, x0, gamma );
console.log( 'x: %d, x0: %d, γ: %d, ln(F(x;x0,γ)): %d', x, x0, gamma, y );
}