Logarithm of Probability Density Function

Evaluate the natural logarithm of the probability density function (PDF) for a Pareto (Type I) distribution.

The probability density function (PDF) for a Pareto (Type I) random variable is

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

Usage

var logpdf = require( '@stdlib/stats/base/dists/pareto-type1/logpdf' );

logpdf( x, alpha, beta )

Evaluates the natural logarithm of the probability density function (PDF) for a Pareto (Type I) distribution with parameters alpha (shape parameter) and beta (scale parameter).

var y = logpdf( 4.0, 1.0, 1.0 );
// returns ~-2.773

y = logpdf( 20.0, 1.0, 10.0 );
// returns ~-3.689

y = logpdf( 7.0, 2.0, 6.0 );
// returns ~-1.561

y = logpdf( 7.0, 6.0, 3.0 );
// returns ~-5.238

y = logpdf( 1.0, 4.0, 2.0 );
// returns -Infinity

y = logpdf( 1.5, 4.0, 2.0 );
// returns -Infinity

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

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

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

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

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

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

y = logpdf( 2.0, 0.0, 0.5 );
// returns NaN

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

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

y = logpdf( 2.0, 0.5, 0.0 );
// returns NaN

logpdf.factory( alpha, beta )

Returns a function for evaluating the natural logarithm of the probability density function (PDF) (CDF) of a Pareto (Type I) distribution with parameters alpha (shape parameter) and beta (scale parameter).

var mylogpdf = logpdf.factory( 0.5, 0.5 );
var y = mylogpdf( 0.8 );
// returns ~-0.705

y = mylogpdf( 2.0 );
// returns ~-2.079

Notes

In virtually all cases, using the logpdf or logcdf functions is preferable to manually computing the logarithm of the pdf or cdf, respectively, since the latter is prone to overflow and underflow.

Examples

var randu = require( '@stdlib/random/base/randu' );
var logpdf = require( '@stdlib/stats/base/dists/pareto-type1/logpdf' );

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

for ( i = 0; i < 10; i++ ) {
    x = randu() * 8.0;
    alpha = randu() * 4.0;
    beta = randu() * 4.0;
    y = logpdf( x, alpha, beta );
    console.log( 'x: %d, α: %d, β: %d, ln(f(x;α,β)): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
}