docs: improve README examples of stats/base/dists/invgamma namespace

PR-URL: #1808
Ref: #1631

Co-authored-by: Philipp Burckhardt <[email protected]>
Co-authored-by: shivam Ahir <[email protected]>
Reviewed-by: Philipp Burckhardt <[email protected]>

Author: ShivamAhir

lib/node_modules/@stdlib/stats/base/dists/invgamma/README.md

@@ -106,10 +106,70 @@ var y = dist.cdf( 0.5 );
 <!-- eslint no-undef: "error" -->
 
 ```javascript
-var objectKeys = require( '@stdlib/utils/keys' );
+var invgammaRandomFactory = require( '@stdlib/random/base/invgamma' ).factory;
+var filledarrayBy = require( '@stdlib/array/filled-by' );
+var variance = require( '@stdlib/stats/base/variance' );
+var linspace = require( '@stdlib/array/base/linspace' );
+var gamma = require( '@stdlib/stats/base/dists/gamma' );
+var mean = require( '@stdlib/stats/base/mean' );
+var abs = require( '@stdlib/math/base/special/abs' );
 var invgamma = require( '@stdlib/stats/base/dists/invgamma' );
 
-console.log( objectKeys( invgamma ) );
+// Define the shape and scale parameters:
+var alpha = 5.0; // shape parameter (α)
+var beta = 1.0;  // scale parameter (β)
+
+// Generate an array of x values:
+var x = linspace( 0.01, 3.0, 100 );
+
+// Compute the PDF for each x:
+var invgammaPDF = invgamma.pdf.factory( alpha, beta );
+var pdf = filledarrayBy( x.length, 'float64', invgammaPDF );
+
+// Compute the CDF for each x:
+var invgammaCDF = invgamma.cdf.factory( alpha, beta );
+var cdf = filledarrayBy( x.length, 'float64', invgammaCDF );
+
+// Output the PDF and CDF values:
+console.log( 'x values:', x );
+console.log( 'PDF values:', pdf );
+console.log( 'CDF values:', cdf );
+
+// Compute statistical properties:
+var theoreticalMean = invgamma.mean( alpha, beta );
+var theoreticalVariance = invgamma.variance( alpha, beta );
+var theoreticalSkewness = invgamma.skewness( alpha, beta );
+var theoreticalKurtosis = invgamma.kurtosis( alpha, beta );
+
+console.log( 'Theoretical Mean:', theoreticalMean );
+console.log( 'Theoretical Variance:', theoreticalVariance );
+console.log( 'Skewness:', theoreticalSkewness );
+console.log( 'Kurtosis:', theoreticalKurtosis );
+
+// Generate random samples from the inverse gamma distribution:
+var rinvGamma = invgammaRandomFactory( alpha, beta );
+var n = 1000;
+var samples = filledarrayBy( n, 'float64', rinvGamma );
+
+// Compute sample mean and variance:
+var sampleMean = mean( n, samples, 1 );
+var sampleVariance = variance( n, 1, samples, 1 );
+
+console.log( 'Sample Mean:', sampleMean );
+console.log( 'Sample Variance:', sampleVariance );
+
+// Compare sample statistics to theoretical values:
+console.log( 'Difference in Mean:', abs( mean - sampleMean ) );
+console.log( 'Difference in Variance:', abs( variance - sampleVariance ) );
+
+// Demonstrate the relationship between inverse gamma and gamma distributions:
+var y = 0.5;
+var invGammaCDF = invgamma.cdf( y, alpha, beta );
+var gammaCDF = 1.0 - gamma.cdf( 1.0 / y, alpha, 1.0 / beta );
+
+console.log( 'Inverse Gamma CDF at y =', y, ':', invGammaCDF );
+console.log( '1 - Gamma CDF at 1/y =', 1 / y, ':', gammaCDF );
+console.log( 'Difference:', abs( invGammaCDF - gammaCDF ) );
 ```
 
 </section>

lib/node_modules/@stdlib/stats/base/dists/invgamma/examples/index.js

@@ -18,7 +18,67 @@
 
 'use strict';
 
-var objectKeys = require( '@stdlib/utils/keys' );
+var invgammaRandomFactory = require( '@stdlib/random/base/invgamma' ).factory;
+var filledarrayBy = require( '@stdlib/array/filled-by' );
+var variance = require( '@stdlib/stats/base/variance' );
+var linspace = require( '@stdlib/array/base/linspace' );
+var gamma = require( '@stdlib/stats/base/dists/gamma' );
+var mean = require( '@stdlib/stats/base/mean' );
+var abs = require( '@stdlib/math/base/special/abs' );
 var invgamma = require( './../lib' );
 
-console.log( objectKeys( invgamma ) );
+// Define the shape and scale parameters:
+var alpha = 5.0; // shape parameter (α)
+var beta = 1.0;  // scale parameter (β)
+
+// Generate an array of x values:
+var x = linspace( 0.01, 3.0, 100 );
+
+// Compute the PDF for each x:
+var invgammaPDF = invgamma.pdf.factory( alpha, beta );
+var pdf = filledarrayBy( x.length, 'float64', invgammaPDF );
+
+// Compute the CDF for each x:
+var invgammaCDF = invgamma.cdf.factory( alpha, beta );
+var cdf = filledarrayBy( x.length, 'float64', invgammaCDF );
+
+// Output the PDF and CDF values:
+console.log( 'x values:', x );
+console.log( 'PDF values:', pdf );
+console.log( 'CDF values:', cdf );
+
+// Compute statistical properties:
+var theoreticalMean = invgamma.mean( alpha, beta );
+var theoreticalVariance = invgamma.variance( alpha, beta );
+var theoreticalSkewness = invgamma.skewness( alpha, beta );
+var theoreticalKurtosis = invgamma.kurtosis( alpha, beta );
+
+console.log( 'Theoretical Mean:', theoreticalMean );
+console.log( 'Theoretical Variance:', theoreticalVariance );
+console.log( 'Skewness:', theoreticalSkewness );
+console.log( 'Kurtosis:', theoreticalKurtosis );
+
+// Generate random samples from the inverse gamma distribution:
+var rinvGamma = invgammaRandomFactory( alpha, beta );
+var n = 1000;
+var samples = filledarrayBy( n, 'float64', rinvGamma );
+
+// Compute sample mean and variance:
+var sampleMean = mean( n, samples, 1 );
+var sampleVariance = variance( n, 1, samples, 1 );
+
+console.log( 'Sample Mean:', sampleMean );
+console.log( 'Sample Variance:', sampleVariance );
+
+// Compare sample statistics to theoretical values:
+console.log( 'Difference in Mean:', abs( mean - sampleMean ) );
+console.log( 'Difference in Variance:', abs( variance - sampleVariance ) );
+
+// Demonstrate the relationship between inverse gamma and gamma distributions:
+var y = 0.5;
+var invGammaCDF = invgamma.cdf( y, alpha, beta );
+var gammaCDF = 1.0 - gamma.cdf( 1.0 / y, alpha, 1.0 / beta );
+
+console.log( 'Inverse Gamma CDF at y =', y, ':', invGammaCDF );
+console.log( '1 - Gamma CDF at 1/y =', 1 / y, ':', gammaCDF );
+console.log( 'Difference:', abs( invGammaCDF - gammaCDF ) );