Random-variate-generation-examples
Random variate generation examples
Introduction
This document is a concise guide for generating random variates with the probability distributions of the Raku package "Statistics::Distributions", [AAp1].
Setup
use Statistics::Distributions;
use Data::Summarizers;
use Text::Plot;# (Any)Univariate Distributions
Beta Distribution
Beta distribution is a continuous probability distribution that is defined by two parameters, alpha and beta. The beta distribution is often used to model proportions.
my $beta = BetaDistribution.new(4, 4);
my @res = random-variate($beta, 200);
records-summary(@res);# +-------------------------------+
# | numerical |
# +-------------------------------+
# | Mean => 0.4933713295636193 |
# | 1st-Qu => 0.3844090947246037 |
# | 3rd-Qu => 0.6167605580528748 |
# | Min => 0.07249604622942857 |
# | Median => 0.4761216744181751 |
# | Max => 0.9689344121753427 |
# +-------------------------------+text-histogram(@res, title => 'Beta Distribution')# Beta Distribution
# +----------+----------+-----------+-----------+----------+-+
# | |
# + ā” ā” + 25.00
# | * * |
# + * * + 20.00
# | * * ā” ā” |
# + * * * ā” * + 15.00
# | ā” ā” * * * ā” * ā” * |
# + * * * * * * * * * ā” + 10.00
# | ā” ā” * * * * * * * * * * |
# + ā” * * * * * * * * * * * ā” * + 5.00
# | * * * * * * * * * * * * * * |
# + ā” * * * * * * * * * * * * * * ā” ā” ā” + 0.00
# | |
# +----------+----------+-----------+-----------+----------+-+
# 0.20 0.40 0.60 0.80 1.00Bernoulli Distribution
Bernoulli distribution is a discrete probability distribution that takes the value 1 with probability p and the value 0 with probability 1-p.
my $bernoulli = BernoulliDistribution.new(:p(0.5));
my @res = random-variate($bernoulli, 12);
records-summary(@res);# +--------------------+
# | numerical |
# +--------------------+
# | Mean => 0.583333 |
# | 1st-Qu => 0 |
# | 3rd-Qu => 1 |
# | Median => 1 |
# | Min => 0 |
# | Max => 1 |
# +--------------------+Discrete Uniform Distribution
Discrete uniform distribution is a discrete probability distribution that takes on a finite number of values with equal probability.
my $discrete_uniform = DiscreteUniformDistribution.new(:min(10), :max(20));
my @res = random-variate($discrete_uniform, 200);
records-summary(@res);# +--------------+
# | numerical |
# +--------------+
# | Max => 20 |
# | 1st-Qu => 12 |
# | Mean => 15 |
# | 3rd-Qu => 18 |
# | Min => 10 |
# | Median => 15 |
# +--------------+text-list-plot(@res.&tally.kv.rotor(2), title => 'Discrete Uniform Distribution tallies')# Discrete Uniform Distribution tallies
# +---+---------+---------+----------+---------+---------+---+
# | |
# | * |
# + * + 25.00
# | |
# | * |
# + + 20.00
# | * |
# | * * * * |
# | |
# + * + 15.00
# | * |
# | * |
# + + 10.00
# +---+---------+---------+----------+---------+---------+---+
# 10.00 12.00 14.00 16.00 18.00 20.00Normal Distribution
Normal distribution is a continuous probability distribution that is defined by two parameters, the mean and the standard deviation. The normal distribution is also known as the Gaussian distribution.
my $normal = NormalDistribution.new(:mean(10), :sd(2));
my @res = random-variate($normal, 200);
records-summary(@res);# +------------------------------+
# | numerical |
# +------------------------------+
# | Min => 4.717058781945211 |
# | Mean => 9.945278573794406 |
# | 3rd-Qu => 11.382490722586429 |
# | Max => 15.066827262353309 |
# | 1st-Qu => 8.63330161702207 |
# | Median => 9.931032883779704 |
# +------------------------------+text-histogram(@res, title => 'Normal Distribution')# Normal Distribution
# +---------+---------+---------+---------+---------+--------+
# | |
# | ā” ā” |
# + * * + 20.00
# | ā” * ā” * ā” |
# + * ā” * ā” * * * + 15.00
# | * * * * * * ā” * ā” |
# | * * * * * * * * * |
# + ā” * * * * * * * * * + 10.00
# | ā” * * * * * * * * * * ā” ā” |
# + ā” * * * * * * * * * * * * * + 5.00
# | ā” ā” ā” * * * * * * * * * * * * * * ā” ā” |
# + * * * * * * * * * * * * * * * * * * ā” * ā” + 0.00
# | |
# +---------+---------+---------+---------+---------+--------+
# 6.00 8.00 10.00 12.00 14.00Uniform Distribution
Uniform distribution is a continuous probability distribution that is defined by two parameters, the minimum and the maximum. The uniform distribution is also known as the rectangular distribution.
my $uniform = UniformDistribution.new(:min(-10), :max(5));
my @res = random-variate($uniform, 200);
records-summary(@res);# +------------------------------+
# | numerical |
# +------------------------------+
# | 1st-Qu => -6.138998899203754 |
# | Min => -9.992336384814417 |
# | 3rd-Qu => 0.9587773392229311 |
# | Median => -2.60046162992163 |
# | Mean => -2.653097771031485 |
# | Max => 4.966623004322626 |
# +------------------------------+text-histogram(@res, title => 'Uniform Distribution')# Uniform Distribution
# +---+----------------+----------------+----------------+---+
# | |
# | ā” ā” ā” ā” ā” |
# | * * ā” ā” ā” * * ā” * |
# + * ā” ā” * * * * * ā” * ā” * * + 10.00
# | * ā” * * * * * * * * * * * * ā” |
# | * * * * * ā” * * * * ā” * * * * * * |
# | * * ā” * * * * * * * * * * * * ā” * * * |
# + * * * * * * * * * * * * * * * * * * * + 5.00
# | * * * * * * * * * * * * * * * * * * * |
# | * * * * * * * * * * * * * * * * * * * ā” |
# | * * * * * * * * * * * * * * * * * * * * |
# + * * * * * * * * * * * * * * * * * * * * ā” + 0.00
# | |
# +---+----------------+----------------+----------------+---+
# -10.00 -5.00 0.00 5.00Multivariate Distributions
Binormal Distribution
Binormal distribution represents a bivariate normal distribution with mean [Ī¼1, Ī¼2]
and covariance matrix [[Ļ1 ** 2, Ļ * Ļ1 * Ļ2], [Ļ * Ļ1 * Ļ2, Ļ2 **2]].
my $binormal = BinormalDistribution.new([10, 4], [4, 2], 0.5);
my @res = random-variate($binormal, 40);
records-summary(@res, field-names => ['0', '1']);# +------------------------------+------------------------------+
# | 0 | 1 |
# +------------------------------+------------------------------+
# | Min => 3.126845329726824 | Min => 0.5669495165957725 |
# | 1st-Qu => 6.796151847845332 | 1st-Qu => 2.944060696125733 |
# | Mean => 9.92700023359102 | Mean => 4.4038168050578985 |
# | Median => 9.96620699701577 | Median => 4.4678447070721745 |
# | 3rd-Qu => 12.097628186652472 | 3rd-Qu => 5.7560359097338685 |
# | Max => 19.607603457253706 | Max => 8.915840675765043 |
# +------------------------------+------------------------------+text-list-plot(@res, width => 60, height => 20, title => 'Binormal Distribution random variates')# Binormal Distribution random variates
# +--------+---------------+---------------+---------------+-+
# | |
# | * |
# | |
# + * + 8.00
# | * * |
# | * * * |
# + * * + 6.00
# | * * * |
# | * * ** * |
# | ** * * |
# + * * + 4.00
# | * ** * * |
# | * * |
# | * * * * |
# + * + 2.00
# | * * |
# | * |
# + + 0.00
# +--------+---------------+---------------+---------------+-+
# 5.00 10.00 15.00 20.00Derived Distributions
Mixture Distribution
Mixture distribution is a probability distribution that is a weighted sum of two or more other probability distributions.
my $mixture = MixtureDistribution.new([2, 5], [NormalDistribution.new(3, 4), NormalDistribution.new(16, 5)]);
my @res = random-variate($mixture, 300);
records-summary(@res);# +------------------------------+
# | numerical |
# +------------------------------+
# | Max => 29.97988072254067 |
# | 1st-Qu => 5.625553940978383 |
# | 3rd-Qu => 17.64796620872335 |
# | Median => 12.818222578426187 |
# | Min => -7.442778967486351 |
# | Mean => 11.707767528338996 |
# +------------------------------+text-histogram(@res, title => 'Mixture Distribution', width => 80)# Mixture Distribution
# +-----------------+------------------+------------------+-----------------+----+
# | |
# + ā” ā” + 30.00
# | ā” * * ā” |
# + * * * * + 25.00
# | * * * * |
# | * * * * |
# + * * * * ā” + 20.00
# | ā” ā” ā” * * * * * |
# | ā” * ā” * * * * * * ā” * |
# + * * * * * * * * * * * + 15.00
# | ā” * * * * * * * * * * * ā” |
# + ā” * * * * * * * * * * * * * + 10.00
# | * * * * * * * * * * * * * * ā” |
# | ā” ā” * * * * * * * * * * * * * * * |
# + ā” * * * * * * * * * * * * * * * * * + 5.00
# | * * * * * * * * * * * * * * * * * * ā” |
# + * * * * * * * * * * * * * * * * * * * ā” + 0.00
# | |
# +-----------------+------------------+------------------+-----------------+----+
# 0.00 10.00 20.00 30.00Product Distribution
Product distribution is a probability distribution that is the product of two or more other probability distributions.
my $product = ProductDistribution.new([NormalDistribution.new(3, 4), NormalDistribution.new(6, 5)]);
my @res = random-variate($product, 60);
records-summary(@res, field-names => ['0', '1']);# +------------------------------+------------------------------+
# | 0 | 1 |
# +------------------------------+------------------------------+
# | Min => -4.16776860721348 | Min => -9.377737554909633 |
# | 1st-Qu => 1.0168330466759363 | 1st-Qu => 2.874748295461403 |
# | Mean => 3.238636694793194 | Mean => 6.092095526592996 |
# | Median => 2.998735983735804 | Median => 6.578421106674605 |
# | 3rd-Qu => 5.86664338890022 | 3rd-Qu => 9.853530032695765 |
# | Max => 12.084239335313105 | Max => 22.529073896534314 |
# +------------------------------+------------------------------+text-list-plot(@res, width => 60, height => 20, title => "Product Distribution random variates")# Product Distribution random variates
# ++---------------+---------------+---------------+---------+
# | |
# | * |
# + + 20.00
# | |
# | * |
# | * * * |
# | * * * * * |
# + * * *** * * * * + 10.00
# | * * * * * * * * * * |
# | * * * *** |
# | * * * * * * * * * * |
# | * * * * |
# + * * * * + 0.00
# | * * |
# | * * |
# | |
# | * |
# + + -10.00
# ++---------------+---------------+---------------+---------+
# -5.00 0.00 5.00 10.00References
[AAp1] Anton Antonov Statistics::Distributions Raku package, (2024), GitHub/antononcube.