# Probability Density Function

Inverse gamma distribution probability density function (PDF).

The probability density function (PDF) for an inverse gamma random variable is

where alpha > 0 is the shape parameter and beta > 0 is the scale parameter.

## Usage

var pdf = require( '@stdlib/math/base/dists/invgamma/pdf' );


#### pdf( x, alpha, beta )

Evaluates the probability density function (PDF) for an inverse gamma distribution with parameters alpha (shape parameter) and beta (rate parameter).

var y = pdf( 2.0, 0.5, 1.0 );
// returns ~0.121

y = pdf( 0.2, 1.0, 1.0 );
// returns ~0.168

y = pdf( -1.0, 4.0, 2.0 );
// returns 0.0


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

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

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

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


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

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

y = pdf( 2.0, -0.5, 1.0 );
// returns NaN


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

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

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


#### pdf.factory( alpha, beta )

Returns a function for evaluating the PDF of an inverse gamma distribution with parameters alpha (shape parameter) and beta (rate parameter).

var myPDF = pdf.factory( 6.0, 7.0 );

var y = myPDF( 2.0 );
// returns ~0.231


## Examples

var randu = require( '@stdlib/random/base/randu' );
var pdf = require( '@stdlib/math/base/dists/invgamma/pdf' );

var alpha;
var beta;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
x = randu() * 2.0;
alpha = randu() * 5.0;
beta = randu() * 5.0;
y = pdf( x, alpha, beta );
console.log( 'x: %d, α: %d, β: %d, f(x;α,β): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
}