README.md

Cumulative Distribution Function

Erlang distribution cumulative distribution function.

The cumulative distribution function for a Erlang random variable is

$$F(x; k,\lambda) = 1 - \sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^n$$

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/stats/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 discreteUniform = require( '@stdlib/random/array/discrete-uniform' );
var uniform = require( '@stdlib/random/array/uniform' );
var logEachMap = require( '@stdlib/console/log-each-map' );
var cdf = require( '@stdlib/stats/base/dists/erlang/cdf' );

var opts = {
    'dtype': 'float64'
};
var x = uniform( 20, 0.0, 10.0, opts );
var k = discreteUniform( 20, 0, 10, opts );
var lambda = uniform( 20, 0.0, 5.0, opts );

logEachMap( 'x: %0.4f, k: %d, λ: %0.4f, F(x;k,λ): %0.4f', x, k, lambda, cdf );

---

C APIs

Usage

#include "stdlib/stats/base/dists/erlang/cdf.h"

stdlib_base_dists_erlang_cdf( x, k, lambda )

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

double y = stdlib_base_dists_erlang_cdf( 2.0, 1, 1.0 );
// returns ~0.865

The function accepts the following arguments:

• x: [in] double input value.
• k: [in] int32_t shape parameter.
• lambda: [in] double rate parameter.

double stdlib_base_dists_erlang_cdf( const double x, const int32_t k, const double lambda );

Examples

#include "stdlib/stats/base/dists/erlang/cdf.h"
#include "stdlib/math/base/special/round.h"
#include <stdint.h>
#include <stdlib.h>
#include <stdio.h>

static double random_uniform( const double min, const double max ) {
    double v = (double)rand() / ( (double)RAND_MAX + 1.0 );
    return min + ( v*(max-min) );
}

int main( void ) {
    double lamda;
    double x;
    int32_t k;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        x = random_uniform( 0.0, 10.0 );
        k = stdlib_base_round( random_uniform( 0.0, 10.0 ) );
        lamda = random_uniform( 0.0, 5.0 );
        y = stdlib_base_dists_erlang_cdf( x, k, lamda );
        printf( "x: %lf, k: %d, lambda: %lf, F(x;k,lambda): %lf\n", x, k, lamda, y );
    }
}