7 Random Number Distributions

This chapter describes the functions for generating random variates and computing their probability densities provided by the PLT Scheme Science Collection/

The functions described in this chapter are defined in the random-distributions sub-collection of the science collection. All of the modules in the random-distributions sub-collection can be made available using the form:

(require williams/science/random-distributions)

The random distribution graphics are provided as separate modules. To also include the random distribution graphics routines, use the following form:

(require williams/science/random-distributions-with-graphics)

The individual modules inthe random-distributions sub-collection can also be made available as described in the sections below.

7.1 The Beta Distribution

Beta Distribution from Wolfram MathWorld.

The beta distribution functions are defined in the "beta.ss" file in the random-distributions subcollection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/beta))

7.1.1 Random Variates from the Beta Distribution

Example: Histogram of random variates from the beta distribution with parameters a = 2.0 and b = 3.0.

#lang scheme

(require (planet williams/science/random-distributions/beta))

(require (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 0.0 1.0)))

(do ((i 0 (+ i 1)))

((= i 10000) (void))

(histogram-increment! h (random-beta 2.0 3.0)))

(histogram-plot h "Histogram of the Beta Distribution"))

The following figure shows the resulting histogram:

7.1.2 Beta Distribution Density Functions

7.1.3 Beta Distribution Graphics

The beta distribution graphics are defined in the "beta-graphics.ss" file in the random-distributions sub-collection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/beta-graphics))

Example: Plot of the probability density and cumulative density of the beta distribution with parameters a = 2.0 and b = 3.0.

#lang scheme

(require (planet williams/science/random-distributions/beta-graphics))

(beta-plot 2.0 3.0)

The following figure shows the resulting plot:

7.2 The Bivariate Gaussian Distribution

Bivariate Normal Distribution from Wolfram MathWorld.

The bivariate Gaussian distribution functions are defined in the "bivariate-gaussian.ss" file in the random-distributions subcollection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/bivariate-gaussian))

7.2.1 Random Variates from the Bivariate Haussian Distribution

Example: 2D histogram of random variates from the bivariate Gaussian distribution with standard deviation 1.0 in both the x and y direction and correlation coefficient 0.0.

#lang scheme

(require (planet williams/science/random-distributions/bivariate))

(require (planet williams/science/histogram-2d-with-graphics))

(let ((h (make-histogram-2d-with-ranges-uniform

20 20 -3.0 3.0 -3.0 3.0)))

(do ((i 0 (+ i 1)))

((= i 10000) (void))

(let-values (((x y) (random-bivariate-gaussian 1.0 1.0 0.0)))

(histogram-2d-increment! h x y)))

(histogram-2d-plot h "Histogram of the Bivariate Gaussian Distribution"))

The following figure shows the resulting histogram:

7.2.2 Bivariate Gaussian Distribution Density Functions

7.2.3 Bivariate Gaussian Distribution Graphics

The bivariate Gaussian distribution graphics are defined in the "bivariate-gaussian-graphics.ss" file in the random-distributions sub-collection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/bivariate-gaussian-graphics))

Example: Plot of the probability density and cumulative density of the bivariate Gaussian distribution mean 0, correlation coefficient 0.0, and standard deviations 1.0 and 1.0 in the x and y directions.

#lang scheme

(require (planet williams/science/random-distributions/binomial-gaussian-graphics))

(bivariate-gaussian-plot 1.0 1.0 0.0)

The following figure shows the resulting plot:

7.3 The Chi-Squared Distribution

Chi-Squared Distribution from Wolfram MathWorld.

The chi-squared distribution functions are defined in the "chi-squared.ss" file in the random-distributions subcollection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/chi-squared))

7.3.1 Random Variates from the Chi-Squared Distribution

Example: Histogram of random variates from the chi-squared distribution with 3.0 degrees of freedom.

#lang scheme

(require (planet williams/science/random-distributions/chi-squared))

(require (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 0.0 10.0)))

(do ((i 0 (+ i 1)))

((= i 10000) (void))

(histogram-increment! h (random-chi-squared 3.0)))

(histogram-plot h "Histogram of the Chi-Squared Distribution"))

The following figure shows the resulting histogram:

7.3.2 Chi-Squared Distribution Density Functions

7.3.3 Chi-Squared Distribution Graphics

The chi-squared distribution graphics are defined in the "chi-squared-graphics.ss" file in the random-distributions sub-collection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/chi-squared-graphics))

Example: Plot of the probability density and cumulative density of the chi-squared distribution with 3.0 degrees of freedom.

#lang scheme

(require (planet williams/science/random-distributions/chi-squared-graphics))

(chi-squared-plot 3.0)

The following figure shows the resulting plot:

7.4 The Exponential Distribution

Exponential Distribution from Wolfram MathWorld.

The exponential distribution functions are defined in the "exponential.ss" file in the random-distributions subcollection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/exponential))

7.4.1 Random Variates from the Exponential Distribution

Example: Histogram of random variates from the exponential distribution with mean 1.0.

#lang scheme

(require (planet williams/science/random-distributions/exponential))

(require (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 0.0 8.0)))

(do ((i 0 (+ i 1)))

((= i 10000) (void))

(histogram-increment! h (random-exponential 1.0)))

(histogram-plot h "Histogram of the Exponential Distribution"))

The following figure shows the resulting histogram:

7.4.2 Exponential Distribution Density Functions

7.4.3 Exponential Distribution Graphics

The exponential distribution graphics are defined in the "exponential-graphics.ss" file in the random-distributions sub-collection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/exponential-graphics))

Example: Plot of the probability density and cumulative density of the exponential distribution with mean 3.0.

#lang scheme

(require (planet williams/science/random-distributions/exponential-graphics))

(exponential-plot 3.0)

The following figure shows the resulting plot:

7.5 The F-Distribution

F-Distribution from Wolfram MathWorld.

The F-distribution functions are defined in the "f-distribution.ss" file in the random-distributions subcollection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/f-distribution))

7.5.1 Random Variates from the F-Distribution

Example: Histogram of random variates from the F-distribution with 2.0 and 3.0 degrees of freedom.

#lang scheme

(require (planet williams/science/random-distributions/f-distribution))

(require (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 0.0 10.0)))

(do ((i 0 (+ i 1)))

((= i 10000) (void))

(histogram-increment! h (random-f-distribution 2.0 3.0)))

(histogram-plot h "Histogram of the F-Distribution"))

The following figure shows the resulting histogram:

7.5.2 F-Distribution Density Functions

7.5.3 F-Distribution Graphics

The F-distribution graphics are defined in the "f-distribution-graphics.ss" file in the random-distributions sub-collection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/f-distribution-graphics))

Example: Plot of the probability density and cumulative density of the Fdistribution with 2.0 and 3.0 degrees of freedom.

#lang scheme

(require (planet williams/science/random-distributions/f-distribution-graphics))

(f-distribution-plot 2.0 3.0)

The following figure shows the resulting plot:

7.6 The Flat (Uniform) Distribution

Uniform Distribution from Wolfram MathWorld.

The flat (uniform) distribution functions are defined in the "flat.ss" file in the random-distributions subcollection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/flat))

Note that the name flat is used because uniform is already used for the more primitive random number functions in SRFI 27. Note that also matches the convention in the GNU Scientific Library [GSL].

7.6.1 Random Variates from the Flat (Uniform) Distribution

Example: Histogram of random variates from the flat (uniform) distribution from 1.0 to 4.0.

#lang scheme

(require (planet williams/science/random-distributions/flat))

(require (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 1.0 4.0)))

(do ((i 0 (+ i 1)))

((= i 10000) (void))

(histogram-increment! h (random-flat 1.0 4.0)))

(histogram-plot h "Histogram of the Flat (Uniform) Distribution"))

The following figure shows the resulting histogram:

7.6.2 Flat (Uniform) Distribution Density Functions

7.6.3 Flat (Uniform) Distribution Graphics

The flat (uniform) distribution graphics are defined in the "flat-graphics.ss" file in the random-distributions sub-collection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/flat-graphics))

Example: Plot of the probability density and cumulative density of the flat (uniform) distribution from 1.0 to 4.0.

#lang scheme

(require (planet williams/science/random-distributions/flat-graphics))

(flat-plot 1.0 4.0)

The following figure shows the resulting plot:

7.7 The Gamma Distribution

Gamma Distribution from Wolfram MathWorld.

The gamma distribution functions are defined in the "gamma.ss" file in the random-distributions subcollection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/gamma))

7.7.1 Random Variates from the Gamma Distribution

Example: Histogram of random variates from the gamma distribution with parameters 3.0 and 3.0.

#lang scheme

(require (planet williams/science/random-distributions/gamma))

(require (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 0.0 24.0)))

(do ((i 0 (+ i 1)))

((= i 10000) (void))

(histogram-increment! h (random-gamma 3.0 3.0)))

(histogram-plot h "Histogram of the Gamma Distribution"))

The following figure shows the resulting histogram:

7.7.2 Gamma Distribution Density Functions

7.7.3 Gamma Distribution Graphics

The gamma distribution graphics are defined in the "gamma-graphics.ss" file in the random-distributions sub-collection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/gamma-graphics))

Example: Plot of the probability density and cumulative density of the gamma distribution with parameters 3.0 and 3.0.

#lang scheme

(require (planet williams/science/random-distributions/gamma-graphics))

(gamma-plot 3.0 3.0)

The following figure shows the resulting plot:

7.8 The Gaussian (Normal) Distribution

Normal Distribution from Wolfram MathWorld.

The Gaussian (normal) distribution functions are defined in the "gaussian.ss" file in the random-distributions subcollection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/gaussian))

7.8.1 Random Variates from the Gaussian (Normal) Distribution

Example: Histogram of random variates from the Gaussian (normal) distribution with mean 10.0 and standard deviation 2.0.

#lang scheme

(require (planet williams/science/random-distributions/gaussian))

(require (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 4.0 16.0)))

(do ((i 0 (+ i 1)))

((= i 10000) (void))

(histogram-increment! h (random-gaussian 10.0 2.0)))

(histogram-plot h "Histogram of the Gaussian (Normal) Distribution"))

The following figure shows the resulting histogram:

Example: Histogram of random variates from the unit Gaussian (normal) distribution.

#lang scheme

(require (planet williams/science/random-distributions/gaussian))

(require (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 -3.0 3.0)))

(do ((i 0 (+ i 1)))

((= i 10000) (void))

(histogram-increment! h (random-unit-gaussian)))

(histogram-plot h "Histogram of the Unit Gaussian (Normal) Distribution"))

The following figure shows the resulting histogram:

7.8.2 Gaussian (Normal) Distribution Density Functions

7.8.3 Gaussian (Normal) Distribution Graphics

The Gaussian (normal) distribution graphics are defined in the "gaussian-graphics.ss" file in the random-distributions sub-collection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/gaussian-graphics))

Example: Plot of the probability density and cumulative density of the Gaussian (normal) distribution with parameters mean 10.0 and standard deviation 2.0.

#lang scheme

(require (planet williams/science/random-distributions/gaussian-graphics))

(gaussian-plot 10.0 2.0)

The following figure shows the resulting plot:

Example: Plot of the probability density and cumulative density of the unit Gaussian (normal) distribution.

#lang scheme

(require (planet williams/science/random-distributions/gaussian-graphics))

(unit-gaussian-plot)

The following figure shows the resulting plot:

7.9 The Gaussian Tail Distribution

The Gaussian tail distribution functions are defined in the "gaussian-tail.ss" file in the random-distributions subcollection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/gaussian-tail))

7.9.1 Random Variates from the Gaussian Tail Distribution

Example: Histogram of random variates from the upper tail greater than 16.0 of the Gaussian distribution with mean 10.0 and standard deviation 2.0.

#lang scheme

(require (planet williams/science/random-distributions/gaussian))

(require (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 16.0 22.0)))

(do ((i 0 (+ i 1)))

((= i 10000) (void))

(histogram-increment! h (random-gaussian-tail 16.0 10.0 2.0)))

(histogram-plot h "Histogram of the Gaussian Tail Distribution"))

The following figure shows the resulting histogram:

7.9.2 Gaussian Tail Distribution Density Functions

7.9.3 Gaussian Tail Distribution Graphics

The Gaussian tail distribution graphics are defined in the "gaussian-tail-graphics.ss" file in the random-distributions sub-collection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/gaussian-tail-graphics))

Example: Plot of the probability density and cumulative density of the upper tail greater than 16.0 of the Gaussian distribution with parameters mean 10.0 and standard deviation 2.0.

#lang scheme

(require (planet williams/science/random-distributions/gaussian-tail-graphics))

(gaussian-plot 16.0 10.0 2.0)

The following figure shows the resulting plot:

Example: Plot of the probability density and cumulative density of the upper tail greater than 3.0 of the unit Gaussian (normal) distribution.

#lang scheme

(require (planet williams/science/random-distributions/gaussian-tail-graphics))

(unit-gaussian-tail-plot 3.0)

The following figure shows the resulting plot:

7.10 The Log Normal Distribution

Log Normal Distribution from Wolfram MathWorld.

The log normal distribution functions are defined in the "lognormal.ss" file in the random-distributions subcollection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/lognormal))

7.10.1 Random Variates from the Log Normal Distribution

Example: Histogram of random variates from the log normal distribution with parameters mean 0.0 and standard deviation 1.0.

#lang scheme

(require (planet williams/science/random-distributions/lognormal))

(require (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 0.0 6.0)))

(do ((i 0 (+ i 1)))

((= i 10000) (void))

(histogram-increment! h (random-log-normal 0.0 1.0)))

(histogram-plot h "Histogram of the Log Normal Distribution"))

The following figure shows the resulting histogram:

7.10.2 Log Normal Distribution Density Functions

7.10.3 Log Normal Distribution Graphics

The log normal distribution graphics are defined in the "lognormal-graphics.ss" file in the random-distributions sub-collection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/lognormal-graphics))

Example: Plot of the probability density and cumulative density of the log normal distribution with mean 0.0 and standard deviation 1.0.

#lang scheme

(require (planet williams/science/random-distributions/lognormal-graphics))

(lognormal-plot 0.0 1.0)

The following figure shows the resulting plot:

7.11 The Pareto Distribution

Pareto Distribution from Wolfram MathWorld.

The Pareto distribution functions are defined in the "pareto.ss" file in the random-distributions subcollection of the science collection and are made available using the form:

(require (planet williams/science/random-distributions/pareto))

7.11.1 Random Variates from the Pareto Distribution

Example: Histogram of random variates from the Pareto distribution with parameters a = 2.0 and b = 3.0.

#lang scheme

(require (planet williams/science/random-distributions/pareto))

(require (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 1.0 21.0)))

(do ((i 0 (+ i 1)))

((= i 10000) (void))

(histogram-increment! h (random-pareto 1.0 1.0)))

(histogram-plot h "Histogram of the Pareto Distribution"))