Cumulative Distribution Function

Erlang distribution cumulative distribution function.

The cumulative distribution function for a Erlang random variable is

upper F left-parenthesis x semicolon k comma lamda right-parenthesis equals 1 minus sigma-summation Underscript n equals 0 Overscript k minus 1 Endscripts StartFraction 1 Over n factorial EndFraction e Superscript minus lamda x Baseline left-parenthesis lamda x right-parenthesis Superscript n

where k is the shape parameter and lambda is the rate parameter. The Erlang distribution is a special case of the gamma distribution, as k is constrained to the natural numbers.

Usage

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

cdf( x, k, lambda )

Evaluates the cumulative distribution function (CDF) for an Erlang distribution with parameters k (shape parameter) and lambda (rate parameter).

var y = cdf( 2.0, 1, 1.0 );
// returns ~0.865

y = cdf( 2.0, 3, 1.0 );
// returns ~0.323

y = cdf( -1.0, 2, 2.0 );
// returns 0.0

y = cdf( -Infinity, 4, 2.0 );
// returns 0.0

y = cdf( +Infinity, 4, 2.0 );
// returns 1.0

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

var y = cdf( NaN, 1, 1.0 );
// returns NaN

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

y = cdf( 0.0, 1, NaN );
// returns NaN

If not provided a nonnegative integer for k, the function returns NaN.

var y = cdf( 2.0, -2, 0.5 );
// returns NaN

y = cdf( 2.0, 0.5, 0.5 );
// returns NaN

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

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

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

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

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

var y = cdf( 2.0, 1, 0.0 );
// returns NaN

y = cdf( 2.0, 1, -5.0 );
// returns NaN

cdf.factory( k, lambda )

Returns a function for evaluating the cumulative distribution function for an Erlang distribution with parameters k (shape parameter) and lambda (rate parameter).

var mycdf = cdf.factory( 2, 0.5 );

var y = mycdf( 6.0 );
// returns ~0.801

y = mycdf( 2.0 );
// returns ~0.264

Examples

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

var lambda;
var k;
var x;
var y;
var i;

for ( i = 0; i < 20; i++ ) {
    x = randu() * 10.0;
    k = round( randu() * 10.0 );
    lambda = randu() * 5.0;
    y = cdf( x, k, lambda );
    console.log( 'x: %d, k: %d, λ: %d, F(x;k,λ): %d', x.toFixed( 4 ), k, lambda.toFixed( 4 ), y.toFixed( 4 ) );
}