Programming language: Crystal
License: GNU General Public License v3.0 or later
Tags: Science And Data Analysis    
Latest version: v0.3.0

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Build Status Crystal Shard Docs License

Alea is a collection of utilities to work with most known probability distributions, written in pure Crystal.

Note: This project is in development state and many distributions are still missing, as well as cumulative distribution functions, so keep in mind that breaking changes may occur frequently.

Why Crystal?

Crystal compiles to really fast native code without sacrificing any of the modern programming languages standards providing a nice and clean interface.



Currently Available

  • PRNGs implementations
  • Random sampling (single/double precision)
  • Cumulative Distribution Functions (single/double precision)

Supported Distributions

Distribution Sampling (32 / 64) CDF (32 / 64)
Beta Y Y N N
Chi-Square Y Y Y Y
Exponential Y Y Y Y
F-Snedecor Y Y N N
Gamma Y Y Y Y
Laplace Y Y Y Y
Log-Normal Y Y Y Y
Normal Y Y Y Y
Poisson N Y N Y
T-Student Y Y N N
Uniform Y Y Y Y


  • Distribution and empirical data statistical properties
  • Quantile Functions


  1. Add the dependency to your shard.yml:
    github: nin93/alea
  1. Run shards install

  2. Import the library:

require "alea"



Random is the interface provided to perform sampling:

random = Alea::Random.new
random.normal # => -0.36790519967553736 : Float64

# Append '32' to call the single-precision version
random.normal32 # => 0.19756398 : Float32

It also accepts an initial seed to reproduce the same seemingly random events across runs:

seed = 9377
random = Alea::Random.new(seed)
random.exp # => 0.10203669577353723 : Float64
Unsafe Methods

Plain sampling methods (such as #normal, #gamma32) performs checks over arguments passed to prevent bad data generation or inner exceptions. In order to avoid checks (might be slow in a large data generation) you must use their unsafe version by prepending next_ to them:

random = Alea::Random.new
random.normal(loc: 0, sigma: 0)      # raises Alea::UndefinedError: sigma is 0 or negative.
random.next_normal(loc: 0, sigma: 0) # these might raise internal exceptions.

Timings are definitely comparable, though: see the benchmarks for direct comparisons between these methods.


Random is actually a wrapper over a well defined pseudo-random number generator. The basic generation of integers and floats comes from the underlying engine, more specifically from: #next_u32, returning a random UInt32, and #next_u64, returning a random UInt64. Floats are obtained by ldexp (load exponent) operations upon generated unsigned integers; signed integers are obtained by raw cast.

Currently implemented engines:

  • XSR128 backed by xoroshiro128++ (32/64 bit)
  • XSR256 backed by xoshiro256++ (32/64 bit)

The digits in the class name stand for the storage of their state in bits. Their period is 2^128 -1 for XSR128 and 2^256 -1 for XSR256.

These engines are from the xoshiro (XOR/shift/rotate) collection, designed by Sebastiano Vigna and David Blackman: really fast generators promising exquisite statistical properties as well.

By default, the PRNG in use by Random is XSR128. You can, though, pass the desired engine as an argument to the constructor. Here is an example using XSR256:

random = Alea::Random.new(Alea::XSR256)
random.float # => 0.6533582874035311 : Float64
random.prng  # => Alea::XSR256

# Or seeded as well
random = Alea::Random.new(193, Alea::XSR256)
random.float # => 0.4507930323670787 : Float64
Custom PRNG

All PRNGs in this library inherit from PRNG. You are allowed to build your own custom PRNG by inheriting the above parent class and defining the methods needed by Alea::Random to ensure proper repeatability and sampling, as described in this example.

It is worth noting that in these implementations #next_u32 and #next_u64 depend on different states and thus they are independent from each other, as well as #next_f32 and #next_f64 or #next_i32 and #next_i64. It is still fine, though, if both #next_u32 and #next_u64 rely on the same state, if you want. I choose not to, as it makes state advancements unpredictable.

Cumulative Distribution Functions

CDF is the interface used to calculate the Cumulative Distribution Functions. Given X ~ D and a fixed quantile x, CDFs are defined as the functions that associate x to the probability that the real-valued random X from the distribution D will take a value less or equal to x.

Arguments passed to CDF methods to shape the distributions are analogous to those used for sampling:

Alea::CDF.normal(0.0)                       # => 0.5 : Float64
Alea::CDF.normal(2.0, loc: 1.0, sigma: 0.5) # => 0.9772498680518208 : Float64
Alea::CDF.chisq(5.279, df: 5.0)             # => 0.6172121213841358 : Float64
Alea::CDF.chisq32(5.279, df: 5.0)           # => 0.61721206 : Float32


Documentation is hosted on GitHub Pages.


Fully listed in LICENSE.md:

  • Crystal Random module for uniform sampling
  • NumPy random module for pseudo-random sampling methods
  • JuliaLang random module for ziggurat methods
  • IncGammaBeta.jl for incomplete gamma functions


  1. Fork it (https://github.com/nin93/alea/fork)
  2. Create your feature branch (git checkout -b my-new-feature)
  3. Commit your changes (git commit -am 'Add some feature')
  4. Push to the branch (git push origin my-new-feature)
  5. Create a new Pull Request


*Note that all licence references and agreements mentioned in the alea README section above are relevant to that project's source code only.