Cumulative Distribution Function

Lognormal distribution cumulative distribution function.

The cumulative distribution function for a lognormal random variable is

upper F left-parenthesis x semicolon mu comma sigma right-parenthesis equals one-half plus one-half e r f left-bracket StartFraction ln x minus mu Over StartRoot 2 EndRoot sigma EndFraction right-bracket

where mu is the location parameter and sigma > 0 is the scale parameter. According to the definition, the natural logarithm of a random variable from a lognormal distribution follows a normal distribution.

Usage

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

cdf( x, mu, sigma )

Evaluates the cumulative distribution function (CDF) for a lognormal distribution with parameters mu (location parameter) and sigma (scale parameter).

var y = cdf( 2.0, 0.0, 1.0 );
// returns ~0.756

y = cdf( 5.0, 10.0, 3.0 );
// returns ~0.003

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

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

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

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

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

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

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

cdf.factory( mu, sigma )

Returns a function for evaluating the cumulative distribution function of a lognormal distribution with parameters mu (location parameter) and sigma (scale parameter).

var mycdf = factory( 3.0, 1.5 );

var y = mycdf( 1.0 );
// returns ~0.023

y = mycdf( 4.0 );
// returns ~0.141

Examples

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

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

for ( i = 0; i < 10; i++ ) {
    x = randu() * 10.0;
    mu = (randu() * 10.0) - 5.0;
    sigma = randu() * 20.0;
    y = cdf( x, mu, sigma );
    console.log( 'x: %d, µ: %d, σ: %d, F(x;µ,σ): %d', x.toFixed( 4 ), mu.toFixed( 4 ), sigma.toFixed( 4 ), y.toFixed( 4 ) );
}