Quantile Function

Erlang distribution quantile function.

The quantile function for an Erlang random variable is

upper Q left-parenthesis p semicolon k comma lamda right-parenthesis equals StartFraction 1 Over lamda EndFraction upper P Superscript negative 1 Baseline left-parenthesis p comma k right-parenthesis

for 0 <= p < 1, where k is the shape parameter and lambda is the rate parameter of the distribution. P^{-1} is the inverse of the lower regularized incomplete gamma function. The Erlang distribution is a special case of the gamma distribution, as k is constrained to the natural numbers.

Usage

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

quantile( p, k, lambda )

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

var y = quantile( 0.8, 2, 1.0 );
// returns ~2.994

y = quantile( 0.5, 4, 2.0 );
// returns ~1.836

If provided a probability p outside the interval [0,1], the function returns NaN.

var y = quantile( 1.9, 1, 1.0 );
// returns NaN

y = quantile( -0.1, 1, 1.0 );
// returns NaN

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

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

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

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

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

var y = quantile( 0.4, -1, 1.0 );
// returns NaN

y = quantile( 0.4, 2.5, 1.0 );
// returns NaN

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

var y = quantile( 0.3, 0, 2.0 );
// returns 0.0

y = quantile( 0.9, 0, 2.0 );
// returns 0.0

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

var y = quantile( 0.4, 1, -1.0 );
// returns NaN

quantile.factory( k, lambda )

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

var myquantile = quantile.factory( 2, 2.0 );
var y = myquantile( 0.8 );
// returns ~1.497

y = myquantile( 0.4 );
// returns ~0.688

Examples

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

var lambda;
var k;
var p;
var y;
var i;

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