Type:	Package
Title:	Convolute Probabilistic Distributions
Version:	1.6.2
URL:	https://github.com/johnaponte/convdistr, https://johnaponte.github.io/convdistr/
Description:	Convolute probabilistic distributions using the random generator function of each distribution. A new random number generator function is created that perform the mathematical operation on the individual random samples from the random generator function of each distribution. See the documentation for examples.
License:	GPL (≥ 3)
Encoding:	UTF-8
Suggests:	spelling, testthat, knitr, rmarkdown
Imports:	stats, graphics, pryr, extraDistr, dplyr, tidyr, ggplot2, RColorBrewer, SHELF, MASS, shiny
RoxygenNote:	7.3.1
VignetteBuilder:	knitr
Language:	en-US
NeedsCompilation:	no
Packaged:	2024-05-05 13:03:36 UTC; master
Author:	Aponte John [aut, cre]
Maintainer:	Aponte John <john.j.aponte@gmail.com>
Repository:	CRAN
Date/Publication:	2024-05-05 13:20:02 UTC

convdistr: Convolute Probabilistic Distributions

Description

Convolute probabilistic distributions using the random generator function of each distribution. A new random number generator function is created that perform the mathematical operation on the individual random samples from the random generator function of each distribution. See the documentation for examples.

Author(s)

Maintainer: Aponte John john.j.aponte@gmail.com (ORCID)

See Also

Useful links:

• https://github.com/johnaponte/convdistr

• https://johnaponte.github.io/convdistr/

Factory for a BETA distribution object

Description

Returns an BETA distribution object that produce random numbers from a beta distribution using the rbeta function

Usage

new_BETA(p_shape1, p_shape2, p_dimnames = "rvar")

new_BETA_lci(p_mean, p_lci, p_uci, p_dimnames = "rvar")

new_BETA_lci2(p_mean, p_lci, p_uci, p_dimnames = "rvar")

Arguments

Value

An object of class DISTRIBUTION, BETA

Functions

• new_BETA_lci(): Constructor based on confidence intervals. Preserve expected value.

• new_BETA_lci2(): Constructor based on ML confidence intervals

Note

When using confidence intervals, the shape parameters are obtained using the following formula:

varp = (p_uci-p_lci)/4^2

shape1 = p_mean * (p_mean * (1 - p_mean) / varp - 1)

shape2 =(1 - p_mean) * (p_mean * (1 - p_mean) / varp - 1)

new_BETA_lci2 estimate parameters using maximum likelihood myDistr <- new_BETA_lci2(0.30,0.25,0.35) myDistr$rfunc(10)

Author(s)

John J. Aponte

Examples

myDistr <- new_BETA(1,1)
myDistr$rfunc(10)
myDistr <- new_BETA_lci(0.30,0.25,0.35)
myDistr$rfunc(10)

Factory for a BETABINOMIAL distribution object

Description

Returns an BETABINOMIAL distribution object that produce random numbers from a betabinomial distribution using the rbbinom function

Usage

new_BETABINOMIAL(p_size, p_shape1, p_shape2, p_dimnames = "rvar")

new_BETABINOMIAL_od(p_size, p_mu, p_od, p_dimnames = "rvar")

new_BETABINOMIAL_icc(p_size, p_mu, p_icc, p_dimnames = "rvar")

Arguments

Value

An object of class DISTRIBUTION, BETADISTRIBUION

Functions

• new_BETABINOMIAL_od(): parametrization based on dispersion

• new_BETABINOMIAL_icc(): parametrization based on intra-class correlation

Note

There are several parametrization for the betabinomial distribution. The one based on shape1 and shape2 are parameters alpha and beta of the beta part of the distribution, but it can be parametrized as mu, and od where mu is the expected mean proportion and od is a measure of the overdispersion.

p_mu = p_shape1/(p_shape1 + p_shape2)

p_od = p_shape1 + p_shape2

p_shape1 = p_mu*p_od

p_shape2 <- (1-p_mu)*p_od

Another parametrization is based on mu and the icc where mu is the mean proportion and icc is the intra-class correlation.

p_mu = p_shape1/(p_shape1 + p_shape2)

p_icc = 1/(p_shape1 + p_shape2 + 1)

p_shape1 = p_mu*(1-p_icc)/p_icc

p_shape2 = (1-p_mu)*(1-p_icc)/p_icc

Author(s)

John J. Aponte

Examples

myDistr <- new_BETABINOMIAL(10,1,1)
myDistr$rfunc(10)

Factory for a BINOMIAL distribution object

Description

Returns a BINOMIAL distribution object that produce random numbers from a binomial distribution using the rbinom function

Usage

new_BINOMIAL(p_size, p_prob, p_dimnames = "rvar")

Arguments

Value

An object of class DISTRIBUTION, BINOMIAL

Author(s)

John J. Aponte

Examples

myDistr <- new_BINOMIAL(1000,0.3)
myDistr$rfunc(10)

Make the convolution of two or more DISTRIBUTION objects

Description

The convolution of the simple algebraic operations is made by the operation of individual drawns of the distributions. The DISTRIBUTION objects must have the same dimensions.

Usage

new_CONVOLUTION(listdistr, op, omit_NA = FALSE)

new_SUM(..., omit_NA = FALSE)

## S3 method for class 'DISTRIBUTION'
e1 + e2

new_SUBTRACTION(..., omit_NA = FALSE)

## S3 method for class 'DISTRIBUTION'
e1 - e2

new_MULTIPLICATION(..., omit_NA = FALSE)

## S3 method for class 'DISTRIBUTION'
e1 * e2

new_DIVISION(..., omit_NA = FALSE)

## S3 method for class 'DISTRIBUTION'
e1 / e2

Arguments

Details

If any of the distributions is of class NA (NA_DISTRIBUTION) the result will be a new distribution of class NA unless the omit_NA option is set to TRUE

Value

and object of class CONVOLUTION, DISTRIBUTION

Functions

• new_SUM(): Sum of distributions

• new_SUBTRACTION(): Subtraction for distributions

• new_MULTIPLICATION(): Multiplication for distributions

• new_DIVISION(): DIVISION for distributions

Author(s)

John J. Aponte

Examples

x1 <- new_NORMAL(0,1)
x2 <- new_UNIFORM(1,2)
new_CONVOLUTION(list(x1,x2), `+`)
new_SUM(x1,x2)
x1 + x2
new_SUBTRACTION(x1,x2)
x1 - x2
new_MULTIPLICATION(list(x1,x2))
x1 * x2
new_DIVISION(list(x1,x2))
x1 / x2

Convolution with association of dimensions

Description

In case of different dimensions of the distribution this function perform the operation on the common distributions and add without modifications the other dimensions of the distribution.

Usage

new_CONVOLUTION_assoc(dist1, dist2, op)

new_SUM_assoc(dist1, dist2)

new_SUBTRACTION_assoc(dist1, dist2)

new_MULTIPLICATION_assoc(dist1, dist2)

new_DIVISION_assoc(dist1, dist2)

Arguments

Details

If distribution A have dimensions a and b and distribution B have dimensions b and c, the A + B would produce a distribution with dimensions a, c, b+b,

Value

an object of class DISTRIBUTION

Functions

• new_SUM_assoc(): Sum of distributions

• new_SUBTRACTION_assoc(): Subtraction of distributions

• new_MULTIPLICATION_assoc(): Multiplication of distributions

• new_DIVISION_assoc(): Division of distributions

Author(s)

John J. Aponte

Examples

x1 <- new_MULTINORMAL(c(0,1), matrix(c(1,0.5,0.5,1),ncol=2), p_dimnames = c("A","B"))
x2 <- new_MULTINORMAL(c(10,1), matrix(c(1,0.4,0.4,1),ncol=2), p_dimnames = c("B","C"))
new_CONVOLUTION_assoc(x1,x2, `+`)
new_SUM_assoc(x1,x2)
new_SUBTRACTION_assoc(x1,x2)
new_MULTIPLICATION_assoc(x1,x2)
new_DIVISION_assoc(x1,x2)

Convolution with combination of dimensions

Description

In case of different dimensions of the distribution this function perform the operation on the combination of the distributions of both distribution.

Usage

new_CONVOLUTION_comb(dist1, dist2, op, p_dimnames)

new_SUM_comb(dist1, dist2)

new_SUBTRACTION_comb(dist1, dist2)

new_MULTIPLICATION_comb(dist1, dist2)

new_DIVISION_comb(dist1, dist2)

Arguments

Details

If distribution A have dimensions a and b and distribution B have dimensions b and c, the A + B would produce a distribution with dimensions a_b,a_c,b_b, b_c

Value

an object of class DISTRIBUTION

Functions

• new_SUM_comb(): Sum of distributions

• new_SUBTRACTION_comb(): Subtraction of distributions

• new_MULTIPLICATION_comb(): Multiplication of distributions

• new_DIVISION_comb(): Division of distributions

Note

In case of the same dimensions, only the first combination is taken

Author(s)

John J. Aponte

Examples

x1 <- new_MULTINORMAL(c(0,1), matrix(c(1,0.5,0.5,1),ncol=2), p_dimnames = c("A","B"))
x2 <- new_MULTINORMAL(c(10,1), matrix(c(1,0.4,0.4,1),ncol=2), p_dimnames = c("B","C"))
new_CONVOLUTION_comb(x1,x2, `+`)
new_SUM_comb(x1,x2)
new_SUBTRACTION_comb(x1,x2)
new_MULTIPLICATION_comb(x1,x2)
new_DIVISION_comb(x1,x2)

Factory for a DIRAC distribution object

Description

Returns an DIRAC distribution object that always return the same number, or the same matrix of numbers in case multiple dimensions are setup

Usage

new_DIRAC(p_scalar, p_dimnames = "rvar")

Arguments

Value

An object of class DISTRIBUTION, DIRAC

Author(s)

John J. Aponte

Examples

myDistr <- new_DIRAC(1)
myDistr$rfunc(10)

Factory for a DIRICHLET distribution object

Description

Returns an DIRICHLET distribution object that draw random numbers generated by the function rdirichlet

Usage

new_DIRICHLET(p_alpha, p_dimnames)

Arguments

Details

A name can be provided for the dimensions. Otherwise rvar1, rvar2, ..., rvark will be assigned

Value

An object of class DISTRIBUTION, p_distribution$distribution, TRUNCATED

Author(s)

John J. Aponte

Examples

myDistr <- new_DIRICHLET(c(0.3,0.2,0.5), c("a","b","c"))
myDistr$rfunc(10)

Factory for a DISCRETE distribution object

Description

Returns an DISCRETE distribution object that sample from the vector p_supp of options with probability the vector of probabilities p_prob.

Usage

new_DISCRETE(p_supp, p_prob, p_dimnames = "rvar")

Arguments

Value

An object of class DISTRIBUTION, DISCRETE

Note

If the second argument is missing, all options will be sample with equal probability. If provided, the second argument would add to 1 and must be the same length that the first argument

Author(s)

John J. Aponte

Examples

myDistr <- new_DISCRETE(p_supp=c(1,2,3,4), p_prob=c(0.40,0.30,0.20,0.10))
myDistr$rfunc(10)

DISTRIBUTION class

Description

DISTRIBUTION is a kind of abstract class (or interface) that the specific constructors should implement.

Details

It contains 4 fields

distribution

A character with the name of the distribution implemented

seed

A numerical that is used for details to produce reproducible details of the distribution

oval

Observed value. Is the value expected. It is used as a number for the mathematical operations of the distributions as if they were a simple scalar

rfunc

A function that generate random numbers from the distribution. Its only parameter n is the number of draws of the distribution. It returns a matrix with as many rows as n, and as many columns as the dimensions of the distributions

The DISTRIBUTION objects could support multidimensional distributions for example DIRICHLET. The names of the dimensions should coincides with the names of the oval vector. If only one dimension, the default name is rvar.

It is expected that the rfunc is included in the creation of new distributions by convolution so the environment should be carefully controlled to avoid reference leaking that is possible within the R language. For that reason, rfunc should be created within a restrict_environment function

Once the object is instanced, the fields are immutable and should not be changed. If the seed needs to be modified, a new object can be created using the set_seed function

Objects are defined for the following distributions

• UNIFORM

• NORMAL

• BETA

• TRIANGULAR

• POISSON

• EXPONENTIAL

• DISCRETE

• DIRAC

• DIRICHLET

• TRUNCATED

• NA_DISTRIBUTION

Value

a DISTRIBUTION object

Author(s)

John J. Aponte

A factory of DISTRIBUTION classes

Description

Generate a function that creates DISTRIBUTION objects

Usage

DISTRIBUTION_factory(distname, rfunction, ovalfunc)

Arguments

Value

A function that is able to create DISTRIBUTION objects.

Note

The function return a new function, that have as arguments the formals of the rfunction plus a new argument dimnames for the dimension names. If The distribution is unidimensional, the default value dimnames = "rvar" will works well, but if not, the dimnames argument should be specified when the generated function is used as in the example for the new_MyDIRICHLET

Author(s)

John J. Aponte

Examples

new_MYDISTR <- DISTRIBUTION_factory("MYDISTR", rnorm, function(){mean})
d1 <- new_MYDISTR(0,1)
summary(d1)
require(extraDistr)
new_MyDIRICHLET <- DISTRIBUTION_factory('rdirichlet',
                       rdirichlet,
                       function() {
                         salpha = sum(alpha)
                         alpha / salpha
                       })
d2 <- new_MyDIRICHLET(c(10, 20, 70), dimnames = c("A", "B", "C"))
summary(d2)

Factory for a EXPONENTIAL distribution using confidence intervals

Description

Returns an EXPONENTIAL distribution object that produce random numbers from an exponential distribution using the rexp function

Usage

new_EXPONENTIAL(p_rate, p_dimnames = "rvar")

Arguments

Value

An object of class DISTRIBUTION, EXPONENTIAL

Author(s)

John J. Aponte

Examples

myDistr <- new_EXPONENTIAL(5)
myDistr$rfunc(10)

Factory for a LOGNORMAL distribution object

Description

Returns a LOGNORMAL distribution object that produce random numbers from a log normal distribution using the rlnorm function

Usage

new_LOGNORMAL(p_meanlog, p_sdlog, p_dimnames = "rvar")

Arguments

Value

An object of class DISTRIBUTION, LOGNORMAL

Author(s)

John J. Aponte

Examples

myDistr <- new_LOGNORMAL(0,1)
myDistr$rfunc(10)

Factory for a NA distribution object

Description

Returns an NA distribution object that always return NA_real_ This is useful to handle NA. By default only one dimension rvar is produced, but if several names are provided more columns will be added to the return matrix

Usage

new_NA(p_dimnames = "rvar")

Arguments

Value

An object of class DISTRIBUTION, NA

Author(s)

John J. Aponte

Examples

myDistr <- new_NA(p_dimnames = "rvar")
myDistr$rfunc(10)

Factory for a NORMAL distribution object

Description

Returns a NORMAL distribution object that produce random numbers from a normal distribution using the rnorm function

Usage

new_NORMAL(p_mean, p_sd, p_dimnames = "rvar")

Arguments

Value

An object of class DISTRIBUTION, NORMAL

Author(s)

John J. Aponte

Examples

myDistr <- new_NORMAL(0,1)
myDistr$rfunc(10)

Factory for a POISSON distribution using confidence intervals

Description

Returns an POISSON distribution object that produce random numbers from a Poisson distribution using the rpois function

Usage

new_POISSON(p_lambda, p_dimnames = "rvar")

Arguments

Value

An object of class DISTRIBUTION, POISSON

Author(s)

John J. Aponte

Examples

myDistr <- new_POISSON(5)
myDistr$rfunc(10)

Factory for a TRIANGULAR distribution object

Description

Returns an TRIANGULAR distribution object that produce random numbers from a triangular distribution using the rtriang function

Usage

new_TRIANGULAR(p_min, p_max, p_mode, p_dimnames = "rvar")

Arguments

Value

An object of class DISTRIBUTION, TRIANGULAR

Author(s)

John J. Aponte

Examples

myDistr <- new_TRIANGULAR(-1,1,0)
myDistr$rfunc(10)

Factory for a TRUNCATED distribution object

Description

Returns an TRUNCATED distribution object that limits the values that are generated by the distribution to be in the limits p_min, p_max

Usage

new_TRUNCATED(p_distribution, p_min = -Inf, p_max = Inf)

Arguments

Value

An object of class DISTRIBUTION, p_distribution$distribution, TRUNCATED

Note

The expected value of a truncated distribution could be very different from the expected value of the unrestricted distribution. Be careful as the oval field is not changed and may not represent any more the expected value of the distribution.

If the distribution is multidimensional, the limits will apply to all dimensions.

Author(s)

John J. Aponte

Examples

myDistr <- new_TRUNCATED(p_distribution = new_NORMAL(0,1), p_min = -1, p_max = 1)
myDistr$rfunc(10)

Factory for a UNIFORM distribution object

Description

Returns an UNIFORM distribution object that produce random numbers from a uniform distribution using the runif function

Usage

new_UNIFORM(p_min, p_max, p_dimnames = "rvar")

Arguments

Value

An object of class DISTRIBUTION, UNIFORM

Author(s)

John J. Aponte

Examples

myDistr <- new_UNIFORM(0,1)
myDistr$rfunc(10)

Adds a total dimension

Description

This function returns a DISTRIBUTION with a new dimension created by row sum of the dimensions of the distribution.

Usage

add_total(p_distribution, p_totalname = "TOTAL")

Arguments

Details

Only works with multidimensional distributions.

Value

a DISTRIBUTION

Author(s)

John J. Aponte

Examples

d1 <- new_DIRICHLET(c(0.2,0.5,0.3))
d2 <- add_total(d1)

cinqnum

Description

Produce 5 numbers of the distribution (mean_, sd_, lci_, uci_, median_).

Usage

cinqnum(x, ...)

## S3 method for class 'DISTRIBUTION'
cinqnum(x, n, ...)

## S3 method for class ''NA''
cinqnum(x, n, ...)

## S3 method for class 'DIRAC'
cinqnum(x, n, ...)

Arguments

Details

Uses the stored seed to have the same sequence always and produce the same numbers This is an internal function for the summary function

Value

a vector with the mean, sd, lci, uci and median values

Methods (by class)

• cinqnum(DISTRIBUTION): Generic method for a DISTRIBUTION

• cinqnum(`NA`): Generic method for optimized for a NA distribution

• cinqnum(DIRAC): Generic method optimized for a DIRAC distribution

Author(s)

John J. Aponte

Fits a beta distribution based on quantiles

Description

Fits a beta distribution based on quantiles

Usage

fitbeta_ml(point, lci, uci)

fitbeta(point, lci, uci)

Arguments

Value

parameters shape1 and shape2 of a beta distribution

Functions

• fitbeta_ml(): using ML to estimate parameters

• fitbeta(): preserve the expected value

Note

This is a wrap of the fitdist to obtain the best parameters for a beta distribution based on quantiles.

When using confidence intervals (not ML), the shape parameters are obtained using the following formula:

varp = (p_uci-p_lci)/4^2

shape1 = p_mean * (p_mean * (1 - p_mean) / varp - 1)

shape2 =(1 - p_mean) * (p_mean * (1 - p_mean) / varp - 1)

Author(s)

John J. Aponte

See Also

fitdist

Examples

fitbeta_ml(0.45,0.40,0.50)
fitbeta(0.45,0.40,0.50)

Fits a Dirichlet distribution,

Description

Fits a Dirichlet distribution based on the parameters of Beta distributions

Usage

fitdirichlet(..., plotBeta = FALSE, n.fitted = "opt")

Arguments

Details

Each one of the arguments is a named vector with values for shape1, shape2. Values from fitbeta are suitable for this. This is a wrap of fitDirichlet

Value

a vector with the parameters for a Dirichlet distribution

Author(s)

John J. Aponte

See Also

fitDirichlet

Examples

a <- fitbeta(0.3, 0.2, 0.4)
c <- fitbeta(0.2, 0.1, 0.3)
b <- fitbeta(0.5, 0.4, 0.6)
fitdirichlet(cat1=a,cat2=b,cat3=c)

Plot of DISTRIBUTION objects using ggplot2

Description

Plot of DISTRIBUTION objects using ggplot2

Usage

ggDISTRIBUTION(x, n = 10000)

Arguments

Value

a ggplot object with the density of the distribution

Examples

x <- new_NORMAL(0,1)
ggDISTRIBUTION(x)
y <- new_DIRICHLET(c(10,20,70))
ggDISTRIBUTION(x)

Metadata for a DISTRIBUTION

Description

Shows the distribution and the oval values of a DISTRIBUTION object

Usage

metadata(x)

## S3 method for class 'DISTRIBUTION'
metadata(x)

## Default S3 method:
metadata(x)

Arguments

Value

A data.frame with the metadata of the distributions

Methods (by class)

• metadata(DISTRIBUTION): Metadata for DISTRIBUTION objects

• metadata(default): Metadata for other objects

Note

The number of columns depends on the dimensions of the distribution. There will be one column distribution with the name of the distribution and one column for each dimension with the names from the oval field.

Author(s)

John J. Aponte

Mixture of DISTRIBUTION objects

Description

Produce a new distribution that obtain random drawns of the mixture of the DISTRIBUTION objects

Usage

new_MIXTURE(listdistr, mixture)

Arguments

Value

an object of class MIXTURE, DISTRIBUTION

Author(s)

John J. Aponte

Examples

x1 <- new_NORMAL(0,1)
x2 <- new_NORMAL(4,1)
x3 <- new_NORMAL(6,1)
new_MIXTURE(list(x1,x2,x3))

Multivariate Normal Distribution

Description

Return a DISTRIBUTION object that draw random numbers from a multivariate normal distribution using the mvrnorm function.

Usage

new_MULTINORMAL(p_mu, p_sigma, p_dimnames, tol = 1e-06, empirical = FALSE)

Arguments

Value

An object of class DISTRIBUTION, MULTINORMAL

Author(s)

John J. Aponte

See Also

mvrnorm

Examples

msigma <- matrix(c(1,0,0,1), ncol=2)
d1 <- new_MULTINORMAL(c(0,1), msigma)
rfunc(d1, 10)

Omit NA distributions from a list of distributions

Description

Omit NA distributions from a list of distributions

Usage

omit_NA(listdistr)

Arguments

Value

the list without the NA_DISTRIBUTION

Author(s)

John J. Aponte

plot of DISTRIBUTION objects

Description

Plot an histogram of the density of the distribution using random numbers from the distribution

Usage

## S3 method for class 'DISTRIBUTION'
plot(x, n = 10000, ...)

Arguments

Value

No return value. Side effect plot the histogram.

Examples

x <- new_NORMAL(0,1)
plot(x)
y <- new_DIRICHLET(c(10,20,70))
plot(x) 

Build a new function with a smaller environment

Description

As standard feature, R include in the environment of a function all the variables that are available when the function is created. This, however is prompt to leak reference when you have a factory of function and they are created within a list.. it will include all the component of the list in the function environment. To prevent that, the random generator functions are encapsulated with a restricted environment where only the variables that the function requires to work are included

Usage

restrict_environment(f, ...)

Arguments

Value

new function with a restricted environment

Author(s)

John J. Aponte

Examples

a = 0
b = 1
myfunc <- restrict_environment(
 function(n) {
   rnorm(meanvalue, sdvalue)
 },
 meanvalue = a, sdvalue = b)

myfunc(10)
ls(envir=environment(myfunc))

Generate random numbers from a DISTRIBUTION object

Description

This is a generic method that calls the rfunc slot of the object

Usage

rfunc(x, n)

Arguments

Value

a matrix with as many rows as n and as many columns as dimensions have distribution

Author(s)

John J. Aponte

Generic function for a DISTRIBUTION object

Description

Generic function for a DISTRIBUTION object

Usage

## S3 method for class 'DISTRIBUTION'
rfunc(x, n)

Arguments

Value

a matrix with as many rows as n and as many columns as

Author(s)

John J. Aponte

Default function

Description

Default function

Usage

## Default S3 method:
rfunc(x, n)

Arguments

Value

No return value. Raise an error message.

Author(s)

John J. Aponte

Check the dimensions of a list of distributions

Description

Check the dimensions of a list of distributions

Usage

same_dimensions(listdistr)

Arguments

Value

return TRUE if all the dimensions are the same

Modify a the seed of a Distribution object

Description

This create a new DISTRIBUTION object but with the specified seed

Usage

set_seed(distribution, seed)

Arguments

Value

a DISTRIBUTION object of the same class

Author(s)

John J. Aponte

Summary of Distributions

Description

Summary of Distributions

Usage

## S3 method for class 'DISTRIBUTION'
summary(object, n = 10000, ...)

Arguments

Value

A data.frame with as many rows as dimensions had the distribution and with the following columns

• distribution name

• varname name of the dimension

• oval value

• nsample number of random samples

• mean_ mean value of the sample

• sd_ standard deviation of the sample

• lci_ lower 95

• median_ median value of the sample

• uci_ upper 95

Note

The sample uses the seed saved in the object those it will provide the same values fir an n value

Author(s)

John J. Aponte