Cumulative Distribution Function

Binomial distribution cumulative distribution function.

The cumulative distribution function for a binomial random variable is

upper F left-parenthesis x semicolon n comma p right-parenthesis equals sigma-summation Underscript i equals 0 Overscript left floor x right floor Endscripts StartBinomialOrMatrix n Choose i EndBinomialOrMatrix p Superscript i Baseline left-parenthesis 1 minus p right-parenthesis Superscript n minus i

where n is the number of trials and p is the success probability. The CDF can be equivalently expressed as

upper F left-parenthesis x semicolon n comma p right-parenthesis equals upper I Subscript 1 minus p Baseline left-parenthesis n minus x comma x plus 1 right-parenthesis

where I is the lower regularized incomplete beta function.

Usage

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

cdf( x, n, p )

Evaluates the cumulative distribution function for a binomial distribution with number of trials n and success probability p.

var y = cdf( 3.0, 20, 0.2 );
// returns ~0.411

y = cdf( 21.0, 20, 0.2 );
// returns 1.0

y = cdf( 5.0, 10, 0.4 );
// returns ~0.834

y = cdf( 0.0, 10, 0.4 );
// returns ~0.06

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

var y = cdf( NaN, 20, 0.5 );
// returns NaN

y = cdf( 0.0, NaN, 0.5 );
// returns NaN

y = cdf( 0.0, 20, NaN );
// returns NaN

If provided a number of trials n which is not a nonnegative integer, the function returns NaN.

var y = cdf( 2.0, 1.5, 0.5 );
// returns NaN

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

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

var y = cdf( 2.0, 20, -1.0 );
// returns NaN

y = cdf( 2.0, 20, 1.5 );
// returns NaN

cdf.factory( n, p )

Returns a function for evaluating the cumulative distribution function of a binomial distribution with number of trials n and success probability p.

var mycdf = cdf.factory( 10, 0.5 );

var y = mycdf( 3.0 );
// returns ~0.172

y = mycdf( 1.0 );
// returns ~0.011

Examples

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

var i;
var n;
var p;
var x;
var y;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 20.0;
    n = round( randu() * 100.0 );
    p = randu();
    y = cdf( x, n, p );
    console.log( 'x: %d, n: %d, p: %d, F(x;n,p): %d', x.toFixed( 4 ), n, p.toFixed( 4 ), y.toFixed( 4 ) );
}