7 Random Number Distributions

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 in the 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.rkt" 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

procedure
(random-beta s a b) → (real-in 0.0 1.0)
  s : random-source?
  a : real?
  b : real?
(unchecked-random-beta s a b) → (real-in 0.0 1.0)
  s : random-source?
  a : real?
  b : real?
(random-beta a b) → (real-in 0.0 1.0)
  a : real?
  b : real?
(unchecked-random-beta a b) → (real-in 0.0 1.0)
  a : real?
  b : real?

Returns a random variate from the beta distribution with parameters a and b using the random source s or (current-random-source) if s is not provided.

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

#lang racket
(require (planet williams/science/random-distributions/beta)
         (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 0.0 1.0)))
  (for ((i (in-range 10000)))
    (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

procedure
(beta-pdf x a b) → (>=/c 0.0)
  x : real?
  a : real?
  b : real?
(unchecked-beta-pdf x a b) → (>=/c 0.0)
  x : real?
  a : real?
  b : real?

Computes the probability density, p(x), at x for the beta distribution with parameters a and b.

procedure
(beta-cdf x a b) → (real-in 0.0 1.0)
  x : real?
  a : real?
  b : real?
(unchecked-beta-cdf x a b) → (real-in 0.0 1.0)
  x : real?
  a : real?
  b : real?

Computes the cumulative density, d(x), at x for the beta distribution with parameters a and b.

7.1.3 Beta Distribution Graphics

The beta distribution graphics are defined in the "beta-graphics.rkt" 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))

procedure
(beta-plot a b) → any
a : real?
b : real?

Returns a plot of the probability density and cumulative density of the beta distribution with parameters a and b. The plot is produced by the plot collection provided with Racket.

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

#lang racket
(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.rkt" 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

procedure
(random-bivariate-gaussian s
sigma-x
sigma-y
rho) →
real? real?
  s : random-source?
  sigma-x : (>=/c 0.0)
  sigma-y : (>=/c 0.0)
  rho : (real-in -1.0 1.0)
(unchecked-random-bivariate-gaussian s
sigma-x
sigma-y
rho) →
real? real?
  s : random-source?
  sigma-x : (>=/c 0.0)
  sigma-y : (>=/c 0.0)
  rho : (real-in -1.0 1.0)
(random-bivariate-gaussian sigma-x
sigma-y
rho) →
real? real?
  sigma-x : (>=/c 0.0)
  sigma-y : (>=/c 0.0)
  rho : (real-in -1.0 1.0)
(unchecked-random-bivariate-gaussian sigma-x
sigma-y
rho) →
real? real?
  sigma-x : (>=/c 0.0)
  sigma-y : (>=/c 0.0)
  rho : (real-in -1.0 1.0)

Returns a pair of correlated Gaussian variates, with mean 0, correlation coefficient rho, and standard deviations sigma-x and sigma-y in the x and y directions using the random source s or (current-random-source) if s is not provided. The correlation coefficient rho must lie between -1 and 1.

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 racket
(require (planet williams/science/random-distributions/bivariate)
         (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)))
  (for ((i (in-range 10000)))
    (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

procedure
(bivariate-gaussian-pdf x
y
sigma-x
sigma-y
rho) → (>=/c 0.0)
  x : real?
  y : real?
  sigma-x : (>=/c 0.0)
  sigma-y : (>=/c 0.0)
  rho : (real-in -1.0 1.0)
(unchecked-bivariate-gaussian-pdf x
y
sigma-x
sigma-y
rho) → (>=/c 0.0)
  x : real?
  y : real?
  sigma-x : (>=/c 0.0)
  sigma-y : (>=/c 0.0)
  rho : (real-in -1.0 1.0)

Computes the probability density, p(x, y), at (x, y) for the bivariate gaussian distribution with mean 0, correlation coefficient rho, and standard deviations sigma-x and sigma-y in the x and y directions.

7.2.3 Bivariate Gaussian Distribution Graphics

The bivariate Gaussian distribution graphics are defined in the "bivariate-gaussian-graphics.rkt" 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))

procedure
(bivariate-gaussian-plot sigma-x
sigma-y
rho) → any
  sigma-x : (>=/c 0.0)
  sigma-y : (>=/c 0.0)
  rho : (real-in -1.0 1.0)

Returns a plot of the probability density and cumulative density of the bivariate Gaussian distribution with mean 0, correlation coefficient rho, and standard deviations sigma-x and sigma-y in the x and y directions. The plot is produced by the plot collection provided with Racket.

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 racket
(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.rkt" 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

procedure
(random-chi-squared s nu) → (>=/c 0.0)
  s : random-source?
  nu : real?
(unchecked-random-chi-squared s nu) → (>=/c 0.0)
  s : random-source?
  nu : real?
(random-chi-squared nu) → (>=/c 0.0)
  nu : real?
(unchecked-random-chi-squared nu) → (>=/c 0.0)
  nu : real?

Returns a random variate from the chi-squared distribution with nu degrees of freedom using the random source s or (current-random-source) if s is not provided.

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

#lang racket
(require (planet williams/science/random-distributions/chi-squared)
         (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 0.0 10.0)))
  (for ((i (in-range 10000)))
    (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

procedure
(chi-squared-pdf x nu) → (>=/c 0.0)
  x : real?
  nu : real?
(unchecked-chi-squared-pdf x nu) → (>=/c 0.0)
  x : real?
  nu : real?

Computes the probability density, p(x), at x for the chi-squared distribution with nu degrees of freedom.

procedure
(chi-squared-cdf x nu) → (real-in 0.0 1.0)
  x : real?
  nu : real?
(unchecked-chi-squared-cdf x nu) → (real-in 0.0 1.0)
  x : real?
  nu : real?

Computes the cumulative density, d(x), at x for the chi-squared distribution with nu degrees of freedom.

7.3.3 Chi-Squared Distribution Graphics

The chi-squared distribution graphics are defined in the "chi-squared-graphics.rkt" 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))

procedure
(chi-squared-plot nu) → any
nu : real?

Returns a plot of the probability density and cumulative density of the chi-squared distribution with nu degrees of freedom. The plot is produced by the plot collection provided with Racket.

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

#lang racket
(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.rkt" 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

procedure
(random-exponential s mu) → (>=/c 0.0)
  s : random-source?
  mu : (>/c 0.0)
(unchecked-random-exponential s mu) → (>=/c 0.0)
  s : random-source?
  mu : (>/c 0.0)
(random-exponential mu) → (>=/c 0.0)
  mu : (>/c 0.0)
(unchecked-random-exponential mu) → (>=/c 0.0)
  mu : (>/c 0.0)

Returns a random variate from the exponential distribution with mean mu using the random source s or (current-random-source) if s is not provided.

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

#lang racket
(require (planet williams/science/random-distributions/exponential)
         (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 0.0 8.0)))
  (for ((i (in-range 10000)))
    (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

procedure
(exponential-pdf x mu) → (>=/c 0.0)
  x : real?
  mu : (>/c 0.0)
(unchecked-exponential-pdf x mu) → (>=/c 0.0)
  x : real?
  mu : (>/c 0.0)

Computes the probability density, p(x), at x for the exponential distribution with mean mu.

procedure
(exponential-cdf x mu) → (real-in 0.0 1.0)
  x : real?
  mu : (>/c 0.0)
(unchecked-exponential-cdf x mu) → (real-in 0.0 1.0)
  x : real?
  mu : (>/c 0.0)

Computes the cumulative density, d(x), at x for the exponential distribution with mean mu.

7.4.3 Exponential Distribution Graphics

The exponential distribution graphics are defined in the "exponential-graphics.rkt" 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))

procedure
(exponential-plot mu) → any
mu : (>/c 0.0)

Returns a plot of the probability density and cumulative density of the exponential distribution with mean mu. The plot is produced by the plot collection provided with Racket.

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

#lang racket
(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.rkt" 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

procedure
(random-f-distribution s nu1 nu2) → (>=/c 0.0)
  s : random-source?
  nu1 : real?
  nu2 : real?
(unchecked-random-f-distribution s nu1 nu2) → (>=/c 0.0)
  s : random-source?
  nu1 : real?
  nu2 : real?
(random-f-distribution nu1 nu2) → (>=/c 0.0)
  nu1 : real?
  nu2 : real?
(unchecked-random-f-distribution nu1 nu2) → (>=/c 0.0)
  nu1 : real?
  nu2 : real?

Returns a random variate from the F-distribution with nu1 and nu2 degrees of freedom using the random source s or (current-random-source) if s is not provided.

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

#lang racket
(require (planet williams/science/random-distributions/f-distribution)(planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 0.0 10.0)))
  (for ((i (in-range 10000)))
    (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

procedure
(f-distribution-pdf x nu1 nu2) → (>=/c 0.0)
  x : real?
  nu1 : real?
  nu2 : real?
(unchecked-f-distribution-pdf x nu1 nu2) → (>=/c 0.0)
  x : real?
  nu1 : real?
  nu2 : real?

Computes the probability density, p(x), at x for the F-distribution with nu1 and nu2 degrees of freedom.

procedure
(f-distribution-cdf x nu1 nu2) → (real-in 0.0 1.0)
  x : real?
  nu1 : real?
  nu2 : real?
(unchecked-f-distribution-cdf x nu1 nu2) → (real-in 0.0 1.0)
  x : real?
  nu1 : real?
  nu2 : real?

Computes the cumulative density, d(x), at x for the F-distribution with nu1 and nu2 degrees of freedom.

7.5.3 F-Distribution Graphics

The F-distribution graphics are defined in the "f-distribution-graphics.rkt" 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))

procedure
(f-distribution-plot nu1 nu2) → any
nu1 : real?
nu2 : real?

Returns a plot of the probability density and cumulative density of the F-distribution with nu1 and nu2 degrees of freedom. The plot is produced by the plot collection provided with Racket.

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

#lang racket
(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.rkt" 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-RM].

7.6.1 Random Variates from the Flat (Uniform) Distribution

procedure
(random-flat s a b) → real?
  s : random-source?
  a : real?
  b : (>/c a)
(unichecked-random-flat s a b) → real?
  s : random-source?
  a : real?
  b : (>/c a)
(random-flat a b) → real?
  a : real?
  b : (>/c a)
(unchecked-random-flat a b) → real?
  a : real?
  b : (>/c a)

Returns a random variate from the flat (uniform) distribution from a to b using the random source s or (current-random-source) if s is not provided.

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

#lang racket
(require (planet williams/science/random-distributions/flat)
         (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 1.0 4.0)))
  (for ((i (in-range 10000)))
    (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

procedure
(flat-pdf x a b) → (>=/c 0.0)
  x : real?
  a : real?
  b : (>/c a)
(unchecked-flat-pdf x a b) → (>=/c 0.0)
  x : real?
  a : real?
  b : (>/c a)

Computes the probability density, p(x), at x for the flat (uniform) distribution from a to b.

procedure
(flat-cdf x a b) → (real-in 0.0 1.0)
  x : real?
  a : real?
  b : (>/c a)
(unchecked-flat-cdf x a b) → (real-in 0.0 1.0)
  x : real?
  a : real?
  b : (>/c a)

Computes the cumulative density, d(x), at x for the flat (uniform) distribution from a to b.

7.6.3 Flat (Uniform) Distribution Graphics

The flat (uniform) distribution graphics are defined in the "flat-graphics.rkt" 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))

procedure
(flat-plot a b) → any
a : real?
b : (>/c a)

Returns a plot of the probability density and cumulative density of the flat (uniform) distribution from a to b. The plot is produced by the plot collection provided with Racket.

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

#lang racket
(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.rkt" 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

procedure
(random-gamma s a b) → (>=/c 0.0)
  s : random-source?
  a : (>/c 0.0)
  b : real?
(unchecked-random-gamma s a b) → (>=/c 0.0)
  s : random-source?
  a : (>/c 0.0)
  b : real?
(random-gamma a b) → (>=/c 0.0)
  a : (>/c 0.0)
  b : real?
(unchecked-random-gamma a b) → (>=/c 0.0)
  a : (>/c 0.0)
  b : real?

Returns a random variate from the gamma distribution with parameters a and b using the random source s or (current-random-source) if s is not provided.

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

#lang racket
(require (planet williams/science/random-distributions/gamma)(planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 0.0 24.0)))
  (for ((i (in-range 10000)))
    (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

procedure
(gamma-pdf x a b) → (>=/c 0.0)
  x : real?
  a : (>=/c 0.0)
  b : real?
(unchecked-gamma-pdf x a b) → (>=/c 0.0)
  x : real?
  a : (>=/c 0.0)
  b : real?

Computes the probability density, p(x), at x for the gamma distribution with parameters a and b.

procedure
(gamma-cdf x a b) → (real-in 0.0 1.0)
  x : real?
  a : (>=/c 0.0)
  b : real?
(unchecked-gamma-cdf x a b) → (real-in 0.0 1.0)
  x : real?
  a : (>=/c 0.0)
  b : real?

Computes the cumulative density, d(x), at x for the gamma distribution with parameters a and b.

7.7.3 Gamma Distribution Graphics

The gamma distribution graphics are defined in the "gamma-graphics.rkt" 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))

procedure
(gamma-plot a b) → any
a : (>=/c 0.0)
b : real?

Returns a plot of the probability density and cumulative density of the gamma distribution with parameters a and b. The plot is produced by the plot collection provided with Racket.

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

#lang racket
(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.rkt" 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

procedure
(random-gaussian s mu sigma) → real?
  s : random-source?
  mu : real?
  sigma : (>=/c 0.0)
(unchecked-random-gaussian s mu sigma) → real?
  s : random-source?
  mu : real?
  sigma : (>=/c 0.0)
(random-gaussian mu sigma) → real?
  mu : real?
  sigma : (>=/c 0.0)
(unchecked-random-gaussian mu sigma) → real?
  mu : real?
  sigma : (>=/c 0.0)

Returns a random variate from the Gaussian (normal) distribution with mean mu and standard deviation sigma using the random source s or (current-random-source) if s is not provided. This function uses the Box-Mueller algorithm that requires two calls to the random source s.

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

#lang racket
(require (planet williams/science/random-distributions/gaussian)
         (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 4.0 16.0)))
  (for ((i (in-range 10000)))
    (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:

procedure
(random-unit-gaussian s) → real?
s : random-source?
(unchecked-random-unit-gaussian s) → real?
s : random-source?
(random-unit-gaussian) → real?
(unchecked-random-unit-gaussian) → real?

Returns a random variate from the Gaussian (normal) distribution with mean 0.0 and standard deviation 1.0 using the random source s or (current-random-source) if s is not provided. This function uses the Box-Mueller algorithm that requires two calls to the random source s.

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

#lang racket
(require (planet williams/science/random-distributions/gaussian)
         (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 -3.0 3.0)))
  (for ((i (in-range 10000)))
    (histogram-increment! h (random-unit-gaussian)))
  (histogram-plot h "Histogram of the Unit Gaussian (Normal) Distribution"))

The following figure shows the resulting histogram:

procedure
(random-gaussian-ratio-method s mu sigma) → real?
  s : random-source?
  mu : real?
  sigma : (>=/c 0.0)
(unchecked-random-gaussian-ratio-method s
mu
sigma) → real?
  s : random-source?
  mu : real?
  sigma : (>=/c 0.0)
(random-gaussian-ratio-method mu sigma) → real?
  mu : real?
  sigma : (>=/c 0.0)
(unchecked-random-gaussian-ratio-method mu
sigma) → real?
  mu : real?
  sigma : (>=/c 0.0)

procedure
(random-unit-gaussian-ratio-method s) → real?
s : random-source?
(unchecked-random-unit-gaussian-ratio-method s) → real?
s : random-source?
(random-unit-gaussian-ratio-method) → real?
(unchecked-random-unit-gaussian-ratio-method) → real?

7.8.2 Gaussian (Normal) Distribution Density Functions

procedure
(gaussian-pdf x mu sigma) → (>=/c 0.0)
  x : real?
  mu : real?
  sigma : (>=/c 0.0)
(unchecked-gaussian-pdf x mu sigma) → (>=/c 0.0)
  x : real?
  mu : real?
  sigma : (>=/c 0.0)

Computes the probability density, p(x), at x for the Gaussian (normal) distribution with mean mu and standard deviation sigma.

procedure
(gaussian-cdf x mu sigma) → (real-in 0.0 1.0)
  x : real?
  mu : real?
  sigma : (>=/c 0.0)
(unchecked-gaussian-cdf x mu sigma) → (real-in 0.0 1.0)
  x : real?
  mu : real?
  sigma : (>=/c 0.0)

Computes the cumulative density, d(x), at x for the Gaussian (normal) distribution with mean mu and standard deviation sigma.

procedure
(unit-gaussian-pdf x) → (>=/c 0.0)
x : real?
(unchecked-unit-gaussian-pdf x) → (>=/c 0.0)
x : real?

Computes the probability density, p(x), at x for the unit Gaussian (normal) distribution.

procedure
(unit-gaussian-cdf x) → (real-in 0.0 1.0)
x : real?
(unchecked-unit-gaussian-cdf x) → (real-in 0.0 1.0)
x : real?

Computes the cumulative density, d(x), at x for the unit Gaussian (normal) distribution.

7.8.3 Gaussian (Normal) Distribution Graphics

The Gaussian (normal) distribution graphics are defined in the "gaussian-graphics.rkt" 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))

procedure
(gaussian-plot mu sigma) → any
mu : real?
sigma : (>=/c 0.0)

Returns a plot of the probability density and cumulative density of the Gaussian (normal) distribution with mean mu and standard deviation sigma. The plot is produced by the plot collection provided with Racket.

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

(gaussian-plot 10.0 2.0)

The following figure shows the resulting plot:

procedure
(unit-gaussian-plot) → any

Returns a plot of the probability density and cumulative density of the unit Gaussian (normal) distribution. The plot is produced by the plot collection provided with Racket.

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

#lang racket
(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.rkt" 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

procedure
(random-gaussian-tail s a mu sigma) → real?
  s : random-source?
  a : real?
  mu : real?
  sigma : (>=/c 0.0)
(unchecked-random-gaussian-tail s a mu sigma) → real?
  s : random-source?
  a : real?
  mu : real?
  sigma : (>=/c 0.0)
(random-gaussian-tail a mu sigma) → real?
  a : real?
  mu : real?
  sigma : (>=/c 0.0)
(unchecked-random-gaussian-tail a mu sigma) → real?
  a : real?
  mu : real?
  sigma : (>=/c 0.0)

Returns a random variate from the upper tail of the Gaussian distribution with mean mu and standard deviation sigma using the random source s or (current-random-source) if s is not provided. The value returned is larger than the lower limit a, which must be greater than the mean mu.

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

(let ((h (make-histogram-with-ranges-uniform 40 16.0 22.0)))
  (for ((i (in-range 10000)))
    (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:

procedure
(random-unit-gaussian-tail s a) → real?
  s : random-source?
  a : (>/c 0.0)
(unchecked-random-unit-gaussian-tail s a) → real?
  s : random-source?
  a : (>/c 0.0)
(random-unit-gaussian-tail a) → real?
  a : (>/c 0.0)
(unchecked-random-unit-gaussian-tail a) → real?
  a : (>/c 0.0)

Returns a random variate from the upper tail of the Gaussian distribution with mean 0 and standard deviation 1 using the random source s or (current-random-source) if s is not provided. The value returned is larger than the lower limit a, which must be greater than the mean mu.

7.9.2 Gaussian Tail Distribution Density Functions

procedure
(gaussian-tail-pdf x a mu sigma) → (>=/c 0.0)
  x : real?
  a : real?
  mu : real?
  sigma : (>=/c 0.0)
(unchecked-gaussian-tail-pdf x a mu sigma) → (>=/c 0.0)
  x : real?
  a : real?
  mu : real?
  sigma : (>=/c 0.0)

Computes the probability density, p(x), at x for the upper tail greater than a of the Gaussian distribution with mean mu and standard deviation sigma.

procedure
(unit-gaussian-tail-pdf x a) → (>=/c 0.0)
  x : real?
  a : (>/c 0.0)
(unit-gaussian-tail-pdf x a) → (>=/c 0.0)
  x : real?
  a : (>/c 0.0)

Computes the probability density, p(x), at x for the upper tail greater than a of the unit Gaussian distribution.

7.9.3 Gaussian Tail Distribution Graphics

The Gaussian tail distribution graphics are defined in the "gaussian-tail-graphics.rkt" 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))

procedure
(gaussian-tail-plot a mu sigma) → any
  a : real?
  mu : real?
  sigma : (>=/c 0.0)

Returns a plot of the probability density and cumulative density of the upper tail greater than a of the Gaussian distribution with mean mu and standard deviation sigma. The plot is produced by the plot collection provided with Racket.

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

(gaussian-plot 16.0 10.0 2.0)

The following figure shows the resulting plot:

procedure
(unit-gaussian-tail-plot a) → any
a : (>/c 0.0)

Returns a plot of the probability density and cumulative density of the upper tail greater than a of the unit Gaussian distribution. The plot is produced by the plot collection provided with Racket.

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

#lang racket
(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.rkt" 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

procedure
(random-lognormal s mu sigma) → real?
  s : random-source?
  mu : real?
  sigma : (>=/c 0.0)
(unchecked-random-lognormal s mu sigma) → real?
  s : random-source?
  mu : real?
  sigma : (>=/c 0.0)
(random-lognormal mu sigma) → real?
  mu : real?
  sigma : (>=/c 0.0)
(unchecked-random-lognormal mu sigma) → real?
  mu : real?
  sigma : (>=/c 0.0)

Returns a random variate from the log normal distribution with mean mu and standard deviation sigma using the random source s or (current-random-source) if s is not provided.

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

#lang racket
(require (planet williams/science/random-distributions/lognormal)
         (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 0.0 6.0)))
  (for ((i (in-range 10000)))
    (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

procedure
(lognormal-pdf x mu sigma) → (>=/c 0.0)
  x : real?
  mu : real?
  sigma : (>=/c 0.0)
(unchecked-lognormal-pdf x mu sigma) → (>=/c 0.0)
  x : real?
  mu : real?
  sigma : (>=/c 0.0)

Computes the probability density, p(x), at x for the log normal distribution with mean mu and standard deviation sigma.

procedure
(lognormal-cdf x mu sigma) → (real-in 0.0 1.0)
  x : real?
  mu : real?
  sigma : (>=/c 0.0)
(unchecked-lognormal-cdf x mu sigma) → (real-in 0.0 1.0)
  x : real?
  mu : real?
  sigma : (>=/c 0.0)

Computes the cumulative density, d(x), at x for the log normal distribution with mean mu and standard deviation sigma.

7.10.3 Log Normal Distribution Graphics

The log normal distribution graphics are defined in the "lognormal-graphics.rkt" 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))

procedure
(lognormal-plot mu sigma) → any
mu : real?
sigma : (>=/c 0.0)

Returns a plot of the probability density and cumulative density of the log normal distribution with mean mu and standard deviation sigma. The plot is produced by the plot collection provided with Racket.

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

#lang racket
(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.rkt" 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

procedure
(random-pareto s a b) → real?
  s : random-source?
  a : real?
  b : real?
(unchecked-random-pareto s a b) → real?
  s : random-source?
  a : real?
  b : real?
(random-pareto a b) → real?
  a : real?
  b : real?
(unchecked-random-pareto a b) → real?
  a : real?
  b : real?

Returns a random variate from the Pareto distribution with parameters a and b using the random source s or (current-random-source) if s is not provided.

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

#lang racket
(require (planet williams/science/random-distributions/pareto)(planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 1.0 21.0)))
  (for ((i (in-range 10000)))
    (histogram-increment! h (random-pareto 1.0 1.0)))
  (histogram-plot h "Histogram of the Pareto Distribution"))

The following figure shows the resulting histogram:

7.11.2 Pareto Distribution Density Functions

procedure
(pareto-pdf x a b) → (>=/c 0.0)
  x : real?
  a : real?
  b : real?
(unchecked-pareto-pdf x a b) → (>=/c 0.0)
  x : real?
  a : real?
  b : real?

Computes the probability density, p(x), at x for the Pareto distribution with parameters a and b.

procedure
(pareto-cdf x a b) → (real-in 0.0 1.0)
  x : real?
  a : real?
  b : real?
(unchecked-pareto-cdf x a b) → (real-in 0.0 1.0)
  x : real?
  a : real?
  b : real?

Computes the cumulative density, d(x), at x for the Pareto distribution with parameters a and b.

7.11.3 Pareto Distribution Graphics

The Pareto distribution graphics are defined in the "pareto-graphics.rkt" file in the random-distributions sub-collection of the Science Collection and are made available using the form:

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

procedure
(pareto-plot a b) → any
a : real?
b : real?

Returns a plot of the probability density and cumulative density of the Pareto distribution with parameters a and b. The plot is produced by the plot collection provided with Racket.

Example: Plot of the probability density and cumulative density of the Pareto distribution with parameters a = 1.0 and b = 1.0.

#lang racket
(require (planet williams/science/random-distributions/pareto-graphics))

(beta-plot 1.0 1.0)

The following figure shows the resulting plot:

7.12 The t-Distribution

Student’s t-Distribution from Wolfram MathWorld.

The t-distribution functions are defined in the "t-distribution.rkt" file in the random-distributions subcollection of the Science Collection and are made available using the form:

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

7.12.1 Random Variates from the t-Distribution

procedure
(random-t-distribution s nu) → real?
  s : random-source?
  nu : real?
(unchecked-random-t-distribution s nu) → real?
  s : random-source?
  nu : real?
(random-t-distribution nu) → real?
  nu : real?
(unchecked-random-t-distribution nu) → real?
  nu : real?

Returns a random variate from the t-distribution with nu degrees of freedom using the random source s or (current-random-source) if s is not provided.

Example: Histogram of random variates from the t-distribution with 1.0 degrees of freedom.

#lang racket
(require (planet williams/science/random-distributions/t-distribution)
         (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 -6.0 6.0)))
  (for ((i (in-range 10000)))
    (histogram-increment! h (random-t-distribution 1.0)))
  (histogram-plot h "Histogram of the t-Distribution"))

The following figure shows the resulting histogram:

7.12.2 t-Distribution Density Functions

procedure
(t-distribution-pdf x nu) → (>=/c 0.0)
  x : real?
  nu : real?
(unchecked-t-distribution-pdf x nu) → (>=/c 0.0)
  x : real?
  nu : real?

Computes the probability density, p(x), at x for the t-distribution with nu degrees of freedom.

procedure
(t-distribution-cdf x nu) → (real-in 0.0 1.0)
  x : real?
  nu : real?
(unchecked-t-distribution-cdf x nu) → (real-in 0.0 1.0)
  x : real?
  nu : real?

Computes the cumulative density, d(x), at x for the t-distribution with nu degrees of freedom.

7.12.3 t-Distribution Graphics

The t-distribution graphics are defined in the "t-distribution-graphics.rkt" file in the random-distributions sub-collection of the Science Collection and are made available using the form:

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

procedure
(t-distribution-plot nu) → any
nu : real?

Returns a plot of the probability density and cumulative density of the t-distribution with nu degrees of freedom. The plot is produced by the plot collection provided with Racket.

Example: Plot of the probability density and cumulative density of the t-distribution with 1.0 degrees of freedom.

#lang racket
(require (planet williams/science/random-distributions/t-distribution-graphics))

(t-distribution-plot 1.0)

The following figure shows the resulting plot:

7.13 The Triangular Distribution

Triangular Distribution from Wolfram MathWorld.

The triangular distribution functions are defined in the "triangular.rkt" file in the random-distributions subcollection of the Science Collection and are made available using the form:

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

7.13.1 Random Variates from the Triangular Distribution

procedure
(random-triangular s a b c) → real?
  s : random-source?
  a : real?
  b : (>/c a)
  c : (real-in a b)
(unchecked-random-triangular s a b c) → real?
  s : random-source?
  a : real?
  b : (>/c a)
  c : (real-in a b)
(random-triangular a b c) → real?
  a : real?
  b : (>/c a)
  c : (real-in a b)
(unchecked-random-triangular a b c) → real?
  a : real?
  b : (>/c a)
  c : (real-in a b)

Returns a random variate from the triangular distribution with minimum value a, maximum value b, and most likely value c using the random source s or (current-random-source) if s is not provided.

Example: Histogram of random variates from the triangular distribution with minimum value 1.0, maximum value 4.0, and most likely value 2.0.

#lang racket
(require (planet williams/science/random-distributions/triangular)
         (planet williams/science/histogram-with-graphics))

(let ((h (make-histogram-with-ranges-uniform 40 1.0 4.0)))
  (for ((i (in-range 10000)))
    (histogram-increment! h (random-traingular 1.0 4.0 2.0)))
  (histogram-plot h "Histogram of the Triangular Distribution"))

The following figure shows the resulting histogram:

7.13.2 Triangular Distribution Density Functions

procedure
(triangular-pdf x a b c) → (>=/c 0.0)
  x : real?
  a : real?
  b : (>/c a)
  c : (real-in a b)
(unchecked-triangular-pdf x a b c) → (>=/c 0.0)
  x : real?
  a : real?
  b : (>/c a)
  c : (real-in a b)

Computes the probability density, p(x), at x for the triangular distribution with minimum value a, maximum value b, and most likely value c.

procedure
(triangular-cdf x a b c) → (real-in 0.0 1.0)
  x : real?
  a : real?
  b : (>/c a)
  c : (real-in a b)
(unchecked-triangular-cdf x a b c) → (real-in 0.0 1.0)
  x : real?
  a : real?
  b : (>/c a)
  c : (real-in a b)

Computes the cumulative density, d(x), at x for the triangular distribution with minimum value a, maximum value b, and most likely value c.

7.13.3 Triangular Distribution Graphics

The triangular distribution graphics are defined in the "triangular-graphics.rkt" file in the random-distributions sub-collection of the Science Collection and are made available using the form:

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

procedure
(triangular-plot a b c) → any
  a : real?
  b : (>/c a)
  c : (real-in a b)

Returns a plot of the probability density and cumulative density of the triangular distribution with minimum value a, maximum value b, and most likely value c. The plot is produced by the plot collection provided with Racket.

Example: Plot of the probability density and cumulative density of the triangular distribution with minimum value 1.0, maximum value 4.0, and most likely value 2.0.

#lang racket
(require (planet williams/science/random-distributions/triangular-graphics))

(triangular-plot 1.0)

The following figure shows the resulting plot:

7.14 The Bernoulli Distribution

Bernoulli Distribution from Wolfram MathWorld.

The Bernoulli distribution functions are defined in the "bernoulli.rkt" file in the random-distributions subcollection of the Science Collection and are made available using the form:

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

7.14.1 Random Variates from the Bernoulli Distribution

procedure
(random-bernoulli s p) → (integer-in 0 1)
  s : random-source?
  p : (real-in 0.0 1.0)
(unchecked-random-bernoulli s p) → (integer-in 0 1)
  s : random-source?
  p : (real-in 0.0 1.0)
(random-bernoulli p) → (integer-in 0 1)
  p : (real-in 0.0 1.0)
(unchecked-random-bernoulli p) → (integer-in 0 1)
  p : (real-in 0.0 1.0)

Returns a random variate from the Bernoulli distribution with probability p using the random source s or (current-random-source) if s is not provided.

Example: Histogram of random variates from the Bernoulli distribution with probability 0.6.

#lang racket
(require (planet williams/science/random-distributions/bernoulli)
         (planet williams/science/discrete-histogram-with-graphics))

(let ((h (make-discrete-histogram)))
  (for ((i (in-range 10000)))
    (discrete-histogram-increment! h (random-bernoulli 0.6)))
  (histogram-plot h "Histogram of the Bernoulli Distribution"))

The following figure shows the resulting histogram:

7.14.2 Bernoulli Distribution Density Functions

procedure
(bernoulli-pdf k p) → (>=/c 0.0)
  k : integer?
  p : (real-in 0.0 1.0)
(unchecked-bernoulli-pdf k p) → (>=/c 0.0)
  k : integer?
  p : (real-in 0.0 1.0)

Computes the probability density, p(k), at k for the Bernoulli distribution with probability p.

procedure
(bernoulli-cdf k p) → (real-in 0.0 1.0)
  k : integer?
  p : (real-in 0.0 1.0)
(unchecked-bernoulli-cdf k p) → (real-in 0.0 1.0)
  k : integer?
  p : (real-in 0.0 1.0)

Computes the cumulative density, d(k), at k for the Bernoulli distribution with probability p.

7.14.3 Bernoulli Distribution Graphics

The Bernoulli distribution graphics are defined in the "bernoulli-graphics.rkt" file in the random-distributions sub-collection of the Science Collection and are made available using the form:

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

procedure
(bernoulli-plot p) → any
p : (real-in 0.0 1.0)

Returns a plot of the probability density and cumulative density of the Bernoulli distribution with probability p. The plot is produced by the plot collection provided with Racket.

Example: Plot of the probability density and cumulative density of the Bernoulli distribution with probability 0.6.

#lang racket
(require (planet williams/science/random-distributions/bernoulli-graphics))

(bernoulli-plot 0.6)

The following figure shows the resulting plot:

7.15 The Binomial Distribution

Binomial Distribution from Wolfram MathWorld.

The binomial distribution functions are defined in the "binomial.rkt" file in the random-distributions subcollection of the Science Collection and are made available using the form:

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

7.15.1 Random Variates from the Binomial Distribution

procedure
(random-binomial s p n) → natural-number/c
  s : random-source?
  p : (real-in 0.0 1.0)
  n : natural-number/c
(unchecked-random-binomial s p n) → natural-number/c
  s : random-source?
  p : (real-in 0.0 1.0)
  n : natural-number/c
(random-binomial p n) → natural-number/c
  p : (real-in 0.0 1.0)
  n : natural-number/c
(unchecked-random-binomial p n) → natural-number/c
  p : (real-in 0.0 1.0)
  n : natural-number/c

Returns a random variate from the binomial distribution with parameters p and n using the random source s or (current-random-source) if s is not provided.

Example: Histogram of random variates from the binomial distribution with parameters 0.5 and 20.

#lang racket
(require (planet williams/science/random-distributions/binomial)
         (planet williams/science/discrete-histogram-with-graphics))

(let ((h (make-discrete-histogram)))
  (for ((i (in-range 10000)))
    (discrete-histogram-increment! h (random-bernoulli 0.5 20)))
  (histogram-plot h "Histogram of the Binomial Distribution"))

The following figure shows the resulting histogram:

7.15.2 Binomial Distribution Density Functions

procedure
(binomial-pdf k p n) → (>=/c 0.0)
  k : integer?
  p : (real-in 0.0 1.0)
  n : natural-number/c
(unchecked-binomial-pdf k p n) → (>=/c 0.0)
  k : integer?
  p : (real-in 0.0 1.0)
  n : natural-number/c

Computes the probability density, p(k), at k for the binomial distribution with parameters p and n.

7.15.3 Binomial Distribution Graphics

The binomial distribution graphics are defined in the "binomial-graphics.rkt" file in the random-distributions sub-collection of the Science Collection and are made available using the form:

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

procedure
(binomial-plot p n) → any
p : (real-in 0.0 1.0)
n : natural-number/c

Returns a plot of the probability density and cumulative density of the binomial distribution with parameters p and n. The plot is produced by the plot collection provided with Racket.

Example: Plot of the probability density and cumulative density of the binomial distribution with parameters 0.5 and 20.

#lang racket
(require (planet williams/science/random-distributions/binomial-graphics))

(bernoulli-plot 0.5 20)

The following figure shows the resulting plot:

7.16 The Geometric Distribution

Geometric Distribution from Wolfram MathWorld.

The geometric distribution functions are defined in the "geometric.rkt" file in the random-distributions subcollection of the Science Collection and are made available using the form:

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

7.16.1 Random Variates from the Geometric Distribution

procedure
(random-geometric s p) → natural-number/c
  s : random-source?
  p : (real-in 0.0 1.0)
(unchecked-random-geometric s p) → natural-number/c
  s : random-source?
  p : (real-in 0.0 1.0)
(random-geometric p) → natural-number/c
  p : (real-in 0.0 1.0)
(unchecked-random-geometric p) → natural-number/c
  p : (real-in 0.0 1.0)

Returns a random variate from the geometric distribution with probability p using the random source s or (current-random-source) if s is not provided.

Example: Histogram of random variates from the geometric distribution with probability 0.5.

#lang racket
(require (planet williams/science/random-distributions/geometric)
         (planet williams/science/discrete-histogram-with-graphics))

(let ((h (make-discrete-histogram)))
  (for ((i (in-range 10000)))
    (discrete-histogram-increment! h (random-geometric 0.5)))
  (histogram-plot h "Histogram of the Geometric Distribution"))

The following figure shows the resulting histogram:

7.16.2 Geometric Distribution Density Functions

procedure
(geometric-pdf k p) → (>=/c 0.0)
  k : integer?
  p : (real-in 0.0 1.0)
(unchecked-geometric-pdf k p) → (>=/c 0.0)
  k : integer?
  p : (real-in 0.0 1.0)

Computes the probability density, p(k), at k for the geometric distribution with probability p.

7.16.3 Geometric Distribution Graphics

The geometric distribution graphics are defined in the "geometric-graphics.rkt" file in the random-distributions sub-collection of the Science Collection and are made available using the form:

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

procedure
(geometric-plot p) → any
p : (real-in 0.0 1.0)

Returns a plot of the probability density and cumulative density of the geometric distribution with probability p. The plot is produced by the plot collection provided with Racket.

Example: Plot of the probability density and cumulative density of the geometric distribution with probability 0.5.

#lang racket
(require (planet williams/science/random-distributions/geometric-graphics))

(geometric-plot 0.5)

The following figure shows the resulting plot:

7.17 The Logarithmic Distribution

Note that the logarithmic distribution in the GSL, and as implemented in the Science Collection, is a discrete version of the exponential distribution.

The logarithmic distribution functions are defined in the "logarithmic.rkt" file in the random-distributions subcollection of the Science Collection and are made available using the form:

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

7.17.1 Random Variates from the Logarithmic Distribution

procedure
(random-logarithmic s p) → natural-number/c
  s : random-source?
  p : (real-in 0.0 1.0)
(unchecked-random-logarithmic s p) → natural-number/c
  s : random-source?
  p : (real-in 0.0 1.0)
(random-logarithmic p) → natural-number/c
  p : (real-in 0.0 1.0)
(unchecked-random-logarithmic p) → natural-number/c
  p : (real-in 0.0 1.0)

Returns a random variate from the logarithmic distribution with probability p using the random source s or (current-random-source) if s is not provided.

Example: Histogram of random variates from the logarithmic distribution with probability 0.5.

#lang racket
(require (planet williams/science/random-distributions/logarithmic)
         (planet williams/science/discrete-histogram-with-graphics))

(let ((h (make-discrete-histogram)))
  (for ((i (in-range 10000)))
    (discrete-histogram-increment! h (random-logarithmic 0.5)))
  (histogram-plot h "Histogram of the Logarithmic Distribution"))

The following figure shows the resulting histogram:

7.17.2 Logarithmic Distribution Density Functions

procedure
(logarithmic-pdf k p) → (>=/c 0.0)
  k : integer?
  p : (real-in 0.0 1.0)
(unchecked-logarithmic-pdf k p) → (>=/c 0.0)
  k : integer?
  p : (real-in 0.0 1.0)

Computes the probability density, p(k), at k for the logarithmic distribution with probability p.

7.17.3 Logarithmic Distribution Graphics

The logarithmic distribution graphics are defined in the "logarithmic-graphics.rkt" file in the random-distributions sub-collection of the Science Collection and are made available using the form:

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

procedure
(logarithmic-plot p) → any
p : (real-in 0.0 1.0)

Returns a plot of the probability density and cumulative density of the logarithmic distribution with probability p. The plot is produced by the plot collection provided with Racket.

Example: Plot of the probability density and cumulative density of the logarithmic distribution with probability 0.5.

#lang racket
(require (planet williams/science/random-distributions/logarithmic-graphics))

(logarithmic-plot 0.5)

The following figure shows the resulting plot:

7.18 The Poisson Distribution

Poisson Distribution from Wolfram MathWorld.

The Poisson distribution functions are defined in the "poisson.rkt" file in the random-distributions subcollection of the Science Collection and are made available using the form:

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

7.18.1 Random Variates from the Poisson Distribution

procedure
(random-poisson s mu) → natural-number/c
  s : random-source?
  mu : real?
(unchecked-random-poisson s mu) → natural-number/c
  s : random-source?
  mu : real?
(random-poisson mu) → natural-number/c
  mu : real?
(unchecked-random-poisson mu) → natural-number/c
  mu : real?

Returns a random variate from the Poisson distribution with mean mu using the random source s or (current-random-source) if s is not provided.

Example: Histogram of random variates from the Poisson distribution with mean 10.0.

#lang racket
(require (planet williams/science/random-distributions/poisson)(planet williams/science/discrete-histogram-with-graphics))

(let ((h (make-discrete-histogram)))
  (for ((i (in-range 10000)))
    (discrete-histogram-increment! h (random-poisson 10.0)))
  (histogram-plot h "Histogram of the Poisson Distribution"))

The following figure shows the resulting histogram:

7.18.2 Poisson Distribution Density Functions

procedure
(poisson-pdf k mu) → (>=/c 0.0)
  k : integer?
  mu : real?
(unchecked-poisson-pdf k mu) → (>=/c 0.0)
  k : integer?
  mu : real?

Computes the probability density, p(k), at k for the Poisson distribution with mean mu.

7.18.3 Poisson Distribution Graphics

The Poisson distribution graphics are defined in the "poisson-graphics.rkt" file in the random-distributions sub-collection of the Science Collection and are made available using the form:

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

procedure
(poisson-plot mu) → any
mu : real?

Returns a plot of the probability density and cumulative density of the Poisson distribution with mean mu. The plot is produced by the plot collection provided with Racket.

Example: Plot of the probability density and cumulative density of the Poisson distribution with mean 10.0.

#lang racket
(require (planet williams/science/random-distributions/poisson-graphics))

(poisson-plot 10.0)

The following figure shows the resulting plot:

7.19 General Discrete Distributions

The discrete distribution functions are defined in the "discrete.rkt" file in the random-distributions subcollection of the Science Collection and are made available using the form:

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

7.19.1 Creating Discrete Distributions

procedure
(discrete? x) → boolean?
x : any/c

Returns true, #t, if x is a discrete distribution.

procedure
(make-discrete weights) → discrete?
weights : (vector-of real?)

Returns a discrete distribution whose probability density is given by the specified weights. Note that the weights do not have to sum to one.

7.19.2 Random Variates from a Discrete Distribution

procedure
(random-discrete s d) → integer?
  s : random-source?
  d : discrete?
(unchecked-random-discrete s d) → integer?
  s : random-source?
  d : discrete?
(random-discrete d) → integer?
  d : discrete?
(unchecked-random-discrete d) → integer?
  d : discrete?

Returns a random variate from a general discrete distribution d using the random source s or (current-random-source) if s is not provided.

Example: Histogram of random variates from a discrete distribution with weights 0.1, 0.4, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, and 0.1.

#lang racket
(require (planet williams/science/random-distributions/discrete)
         (planet williams/science/discrete-histogram-with-graphics))

(let ((h (make-discrete-histogram))
      (d (make-discrete #(0.1 0.4 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1))))
  (for ((i (in-range 10000)))
    (discrete-histogram-increment! h (random-discrete d)))
  (histogram-plot h "Histogram of a Discrete Distribution"))

The following figure shows the resulting histogram:

7.19.3 Discrete Distribution Density Functions

procedure
(discrete-pdf d k) → (real-in 0.0 1.0)
  d : discrete?
  k : integer?
(unchecked-discrete-pdf d k) → (real-in 0.0 1.0)
  d : discrete?
  k : integer?

Computes the probability density, p(k), at k for a discrete distribution d.

procedure
(discrete-cdf d k) → (real-in 0.0 1.0)
  d : discrete?
  k : integer?
(unchecked-discrete-cdf d k) → (real-in 0.0 1.0)
  d : discrete?
  k : integer?

Computes the cumulative density, d(k), at k for a discrete distribution d.

7.19.4 Discrete Distribution Graphics

The discrete distribution graphics are defined in the "discrete-graphics.rkt" file in the random-distributions sub-collection of the Science Collection and are made available using the form:

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

procedure
(discrete-plot d) → any
d : discrete?

Returns a plot of the probability density and cumulative density of a discrete distribution d. The plot is produced by the plot collection provided with Racket.

Example: Plot of the probability density and cumulative density of a discrete distribution with weights 0.1, 0.4, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, and 0.1.

#lang racket
(require (planet williams/science/random-distributions/discrete-graphics))

(let ((d (make-discrete #(0.1 0.4 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1))))
(discrete-plot d))

The following figure shows the resulting plot:

← prev up next →

1	Introduction
2	Using the Science Collection
3	Error Handling
4	Mathematical Functions
5	Special Functions
6	Random Number Generation
7	Random Number Distributions
8	Statistics
9	Histograms
10	Ordinary Differential Equations
11	Chebyshev Approximations
12	Fast Fourier Transforms
13	Physical Constants
14	Unsafe Operations Utilities
15	GNU Free Documentation License
	Bibliography
	Index

7.1	The Beta Distribution
7.2	The Bivariate Gaussian Distribution
7.3	The Chi-Squared Distribution
7.4	The Exponential Distribution
7.5	The F-Distribution
7.6	The Flat (Uniform) Distribution
7.7	The Gamma Distribution
7.8	The Gaussian (Normal) Distribution
7.9	The Gaussian Tail Distribution
7.10	The Log Normal Distribution
7.11	The Pareto Distribution
7.12	The t-Distribution
7.13	The Triangular Distribution
7.14	The Bernoulli Distribution
7.15	The Binomial Distribution
7.16	The Geometric Distribution
7.17	The Logarithmic Distribution
7.18	The Poisson Distribution
7.19	General Discrete Distributions