Cumulative Distribution Function

Poisson distribution cumulative distribution function.

The cumulative distribution function for a Poisson random variable is

upper F left-parenthesis x semicolon lamda right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column for x less-than-or-equal-to 0 2nd Row 1st Column e Superscript negative lamda Baseline sigma-summation Underscript i equals 0 Overscript left floor x right floor Endscripts StartFraction lamda Superscript i Baseline Over i factorial EndFraction 2nd Column for x greater-than 0 EndLayout

where lambda is the mean parameter. Internally, the module evaluates the CDF by evaluating the upper regularized gamma function at input values lambda and floor( x ) + 1.

Usage

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

cdf( x, lambda )

Evaluates the cumulative distribution function for a Poisson distribution with mean parameter lambda.

var y = cdf( 2.0, 0.5 );
// returns ~0.986

y = cdf( 2.0, 10.0 );
// returns ~0.003

y = cdf( -1.0, 4.0 );
// returns 0.0

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

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

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

If provided lambda < 0, the function returns NaN.

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

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

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

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

y = cdf( 10.0, 0.0 );
// returns 1.0

cdf.factory( lambda )

Returns a function for evaluating the cumulative distribution function of a Poisson distribution with mean parameter lambda.

var mycdf = cdf.factory( 5.0 );
var y = mycdf( 3.0 );
// returns ~0.265

y = mycdf( 8.0 );
// returns ~0.932

Examples

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

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

for ( i = 0; i < 10; i++ ) {
    x = randu() * 10.0;
    lambda = randu() * 10.0;
    y = cdf( x, lambda );
    console.log( 'x: %d, λ: %d, F(x;λ): %d', x.toFixed( 4 ), lambda.toFixed( 4 ), y.toFixed( 4 ) );
}