# Random Numbers

Random number generation in Julia uses the Mersenne Twister library via `MersenneTwister`

objects. Julia has a global RNG, which is used by default. Other RNG types can be plugged in by inheriting the `AbstractRNG`

type; they can then be used to have multiple streams of random numbers. Besides `MersenneTwister`

, Julia also provides the `RandomDevice`

RNG type, which is a wrapper over the OS provided entropy.

Most functions related to random generation accept an optional `AbstractRNG`

object as first argument, which defaults to the global one if not provided. Moreover, some of them accept optionally dimension specifications `dims...`

(which can be given as a tuple) to generate arrays of random values.

A `MersenneTwister`

or `RandomDevice`

RNG can generate uniformly random numbers of the following types: `Float16`

, `Float32`

, `Float64`

, `BigFloat`

, `Bool`

, `Int8`

, `UInt8`

, `Int16`

, `UInt16`

, `Int32`

, `UInt32`

, `Int64`

, `UInt64`

, `Int128`

, `UInt128`

, `BigInt`

(or complex numbers of those types). Random floating point numbers are generated uniformly in $[0, 1)$. As `BigInt`

represents unbounded integers, the interval must be specified (e.g. `rand(big.(1:6))`

).

Additionally, normal and exponential distributions are implemented for some `AbstractFloat`

and `Complex`

types, see `randn`

and `randexp`

for details.

## Random generation functions

`Base.rand`

— Function.`rand([rng=GLOBAL_RNG], [S], [dims...])`

Pick a random element or array of random elements from the set of values specified by `S`

; `S`

can be

- an indexable collection (for example
`1:n`

or`['x','y','z']`

), - an
`AbstractDict`

or`AbstractSet`

object, - a string (considered as a collection of characters), or
- a type: the set of values to pick from is then equivalent to
`typemin(S):typemax(S)`

for integers (this is not applicable to`BigInt`

), and to $[0, 1)$ for floating point numbers;

`S`

defaults to `Float64`

(except when `dims`

is a tuple of integers, in which case `S`

must be specified).

**Examples**

```
julia> rand(Int, 2)
2-element Array{Int64,1}:
1339893410598768192
1575814717733606317
julia> rand(MersenneTwister(0), Dict(1=>2, 3=>4))
1=>2
```

The complexity of `rand(rng, s::Union{AbstractDict,AbstractSet})`

is linear in the length of `s`

, unless an optimized method with constant complexity is available, which is the case for `Dict`

, `Set`

and `BitSet`

. For more than a few calls, use `rand(rng, collect(s))`

instead, or either `rand(rng, Dict(s))`

or `rand(rng, Set(s))`

as appropriate.

`Random.rand!`

— Function.`rand!([rng=GLOBAL_RNG], A, [S=eltype(A)])`

Populate the array `A`

with random values. If `S`

is specified (`S`

can be a type or a collection, cf. `rand`

for details), the values are picked randomly from `S`

. This is equivalent to `copyto!(A, rand(rng, S, size(A)))`

but without allocating a new array.

**Examples**

```
julia> rng = MersenneTwister(1234);
julia> rand!(rng, zeros(5))
5-element Array{Float64,1}:
0.5908446386657102
0.7667970365022592
0.5662374165061859
0.4600853424625171
0.7940257103317943
```

`Random.bitrand`

— Function.`bitrand([rng=GLOBAL_RNG], [dims...])`

Generate a `BitArray`

of random boolean values.

**Examples**

```
julia> rng = MersenneTwister(1234);
julia> bitrand(rng, 10)
10-element BitArray{1}:
false
true
true
true
true
false
true
false
false
true
```

`Base.randn`

— Function.`randn([rng=GLOBAL_RNG], [T=Float64], [dims...])`

Generate a normally-distributed random number of type `T`

with mean 0 and standard deviation 1. Optionally generate an array of normally-distributed random numbers. The `Base`

module currently provides an implementation for the types `Float16`

, `Float32`

, and `Float64`

(the default), and their `Complex`

counterparts. When the type argument is complex, the values are drawn from the circularly symmetric complex normal distribution.

**Examples**

```
julia> rng = MersenneTwister(1234);
julia> randn(rng, ComplexF64)
0.6133070881429037 - 0.6376291670853887im
julia> randn(rng, ComplexF32, (2, 3))
2×3 Array{Complex{Float32},2}:
-0.349649-0.638457im 0.376756-0.192146im -0.396334-0.0136413im
0.611224+1.56403im 0.355204-0.365563im 0.0905552+1.31012im
```

`Random.randn!`

— Function.`randn!([rng=GLOBAL_RNG], A::AbstractArray) -> A`

Fill the array `A`

with normally-distributed (mean 0, standard deviation 1) random numbers. Also see the `rand`

function.

**Examples**

```
julia> rng = MersenneTwister(1234);
julia> randn!(rng, zeros(5))
5-element Array{Float64,1}:
0.8673472019512456
-0.9017438158568171
-0.4944787535042339
-0.9029142938652416
0.8644013132535154
```

`Random.randexp`

— Function.`randexp([rng=GLOBAL_RNG], [T=Float64], [dims...])`

Generate a random number of type `T`

according to the exponential distribution with scale 1. Optionally generate an array of such random numbers. The `Base`

module currently provides an implementation for the types `Float16`

, `Float32`

, and `Float64`

(the default).

**Examples**

```
julia> rng = MersenneTwister(1234);
julia> randexp(rng, Float32)
2.4835055f0
julia> randexp(rng, 3, 3)
3×3 Array{Float64,2}:
1.5167 1.30652 0.344435
0.604436 2.78029 0.418516
0.695867 0.693292 0.643644
```

`Random.randexp!`

— Function.`randexp!([rng=GLOBAL_RNG], A::AbstractArray) -> A`

Fill the array `A`

with random numbers following the exponential distribution (with scale 1).

**Examples**

```
julia> rng = MersenneTwister(1234);
julia> randexp!(rng, zeros(5))
5-element Array{Float64,1}:
2.4835053723904896
1.516703605376473
0.6044364871025417
0.6958665886385867
1.3065196315496677
```

`Random.randstring`

— Function.`randstring([rng=GLOBAL_RNG], [chars], [len=8])`

Create a random string of length `len`

, consisting of characters from `chars`

, which defaults to the set of upper- and lower-case letters and the digits 0-9. The optional `rng`

argument specifies a random number generator, see Random Numbers.

**Examples**

```
julia> Random.seed!(0); randstring()
"0IPrGg0J"
julia> randstring(MersenneTwister(0), 'a':'z', 6)
"aszvqk"
julia> randstring("ACGT")
"TATCGGTC"
```

`chars`

can be any collection of characters, of type `Char`

or `UInt8`

(more efficient), provided `rand`

can randomly pick characters from it.

## Subsequences, permutations and shuffling

`Random.randsubseq`

— Function.`randsubseq(A, p) -> Vector`

Return a vector consisting of a random subsequence of the given array `A`

, where each element of `A`

is included (in order) with independent probability `p`

. (Complexity is linear in `p*length(A)`

, so this function is efficient even if `p`

is small and `A`

is large.) Technically, this process is known as "Bernoulli sampling" of `A`

.

`Random.randsubseq!`

— Function.`randsubseq!(S, A, p)`

Like `randsubseq`

, but the results are stored in `S`

(which is resized as needed).

`Random.randperm`

— Function.`randperm([rng=GLOBAL_RNG,] n::Integer)`

Construct a random permutation of length `n`

. The optional `rng`

argument specifies a random number generator (see Random Numbers). To randomly permute an arbitrary vector, see `shuffle`

or `shuffle!`

.

**Examples**

```
julia> randperm(MersenneTwister(1234), 4)
4-element Array{Int64,1}:
2
1
4
3
```

`Random.randperm!`

— Function.`randperm!([rng=GLOBAL_RNG,] A::Array{<:Integer})`

Construct in `A`

a random permutation of length `length(A)`

. The optional `rng`

argument specifies a random number generator (see Random Numbers). To randomly permute an arbitrary vector, see `shuffle`

or `shuffle!`

.

**Examples**

```
julia> randperm!(MersenneTwister(1234), Vector{Int}(undef, 4))
4-element Array{Int64,1}:
2
1
4
3
```

`Random.randcycle`

— Function.`randcycle([rng=GLOBAL_RNG,] n::Integer)`

Construct a random cyclic permutation of length `n`

. The optional `rng`

argument specifies a random number generator, see Random Numbers.

**Examples**

```
julia> randcycle(MersenneTwister(1234), 6)
6-element Array{Int64,1}:
3
5
4
6
1
2
```

`Random.randcycle!`

— Function.`randcycle!([rng=GLOBAL_RNG,] A::Array{<:Integer})`

Construct in `A`

a random cyclic permutation of length `length(A)`

. The optional `rng`

argument specifies a random number generator, see Random Numbers.

**Examples**

```
julia> randcycle!(MersenneTwister(1234), Vector{Int}(undef, 6))
6-element Array{Int64,1}:
3
5
4
6
1
2
```

`Random.shuffle`

— Function.`shuffle([rng=GLOBAL_RNG,] v::AbstractArray)`

Return a randomly permuted copy of `v`

. The optional `rng`

argument specifies a random number generator (see Random Numbers). To permute `v`

in-place, see `shuffle!`

. To obtain randomly permuted indices, see `randperm`

.

**Examples**

```
julia> rng = MersenneTwister(1234);
julia> shuffle(rng, Vector(1:10))
10-element Array{Int64,1}:
6
1
10
2
3
9
5
7
4
8
```

`Random.shuffle!`

— Function.`shuffle!([rng=GLOBAL_RNG,] v::AbstractArray)`

In-place version of `shuffle`

: randomly permute `v`

in-place, optionally supplying the random-number generator `rng`

.

**Examples**

```
julia> rng = MersenneTwister(1234);
julia> shuffle!(rng, Vector(1:16))
16-element Array{Int64,1}:
2
15
5
14
1
9
10
6
11
3
16
7
4
12
8
13
```

## Generators (creation and seeding)

`Random.seed!`

— Function.```
seed!([rng=GLOBAL_RNG], seed) -> rng
seed!([rng=GLOBAL_RNG]) -> rng
```

Reseed the random number generator: `rng`

will give a reproducible sequence of numbers if and only if a `seed`

is provided. Some RNGs don't accept a seed, like `RandomDevice`

. After the call to `seed!`

, `rng`

is equivalent to a newly created object initialized with the same seed.

**Examples**

```
julia> Random.seed!(1234);
julia> x1 = rand(2)
2-element Array{Float64,1}:
0.590845
0.766797
julia> Random.seed!(1234);
julia> x2 = rand(2)
2-element Array{Float64,1}:
0.590845
0.766797
julia> x1 == x2
true
julia> rng = MersenneTwister(1234); rand(rng, 2) == x1
true
julia> MersenneTwister(1) == Random.seed!(rng, 1)
true
julia> rand(Random.seed!(rng), Bool) # not reproducible
true
julia> rand(Random.seed!(rng), Bool)
false
julia> rand(MersenneTwister(), Bool) # not reproducible either
true
```

`Random.MersenneTwister`

— Type.```
MersenneTwister(seed)
MersenneTwister()
```

Create a `MersenneTwister`

RNG object. Different RNG objects can have their own seeds, which may be useful for generating different streams of random numbers. The `seed`

may be a non-negative integer or a vector of `UInt32`

integers. If no seed is provided, a randomly generated one is created (using entropy from the system). See the `seed!`

function for reseeding an already existing `MersenneTwister`

object.

**Examples**

```
julia> rng = MersenneTwister(1234);
julia> x1 = rand(rng, 2)
2-element Array{Float64,1}:
0.5908446386657102
0.7667970365022592
julia> rng = MersenneTwister(1234);
julia> x2 = rand(rng, 2)
2-element Array{Float64,1}:
0.5908446386657102
0.7667970365022592
julia> x1 == x2
true
```

`Random.RandomDevice`

— Type.`RandomDevice()`

Create a `RandomDevice`

RNG object. Two such objects will always generate different streams of random numbers. The entropy is obtained from the operating system.

## Hooking into the `Random`

API

There are two mostly orthogonal ways to extend `Random`

functionalities:

- generating random values of custom types
- creating new generators

The API for 1) is quite functional, but is relatively recent so it may still have to evolve in subsequent releases of the `Random`

module. For example, it's typically sufficient to implement one `rand`

method in order to have all other usual methods work automatically.

The API for 2) is still rudimentary, and may require more work than strictly necessary from the implementor, in order to support usual types of generated values.

### Generating random values of custom types

There are two categories: generating values from a type (e.g. `rand(Int)`

), or from a collection (e.g. `rand(1:3)`

). The simple cases are explained first, and more advanced usage is presented later. We assume here that the choice of algorithm is independent of the RNG, so we use `AbstractRNG`

in our signatures.

#### Generating values from a type

Given a type `T`

, it's currently assumed that if `rand(T)`

is defined, an object of type `T`

will be produced. In order to define random generation of values of type `T`

, the following method can be defined: `rand(rng::AbstractRNG, ::Random.SamplerType{T})`

(this should return what `rand(rng, T)`

is expected to return).

Let's take the following example: we implement a `Die`

type, with a variable number `n`

of sides, numbered from `1`

to `n`

. We want `rand(Die)`

to produce a die with a random number of up to 20 sides (and at least 4):

```
struct Die
nsides::Int # number of sides
end
Random.rand(rng::AbstractRNG, ::Random.SamplerType{Die}) = Die(rand(rng, 4:20))
# output
```

Scalar and array methods for `Die`

now work as expected:

```
julia> rand(Die)
Die(18)
julia> rand(MersenneTwister(0), Die)
Die(4)
julia> rand(Die, 3)
3-element Array{Die,1}:
Die(6)
Die(11)
Die(5)
julia> a = Vector{Die}(undef, 3); rand!(a)
3-element Array{Die,1}:
Die(18)
Die(6)
Die(8)
```

#### Generating values from a collection

Given a collection type `S`

, it's currently assumed that if `rand(::S)`

is defined, an object of type `eltype(S)`

will be produced. In order to define random generation out of objects of type `S`

, the following method can be defined: `rand(rng::AbstractRNG, sp::Random.SamplerTrivial{S})`

. Here, `sp`

simply wraps an object of type `S`

, which can be accessed via `sp[]`

. Continuing the `Die`

example, we want now to define `rand(d::Die)`

to produce an `Int`

corresponding to one of `d`

's sides:

```
julia> Random.rand(rng::AbstractRNG, d::Random.SamplerTrivial{Die}) = rand(rng, 1:d[].nsides);
julia> rand(Die(4))
3
julia> rand(Die(4), 3)
3-element Array{Any,1}:
3
4
2
```

In the last example, a `Vector{Any}`

is produced; the reason is that `eltype(Die) == Any`

. The remedy is to define `Base.eltype(::Type{Die}) = Int`

.

#### Generating values for an `AbstractFloat`

type

`AbstractFloat`

types are special-cased, because by default random values are not produced in the whole type domain, but rather in `[0,1)`

. The following method should be implemented for `T <: AbstractFloat`

: `Random.rand(::AbstractRNG, ::Random.SamplerTrivial{Random.CloseOpen01{T}})`

#### Optimizing generation with cached computation between calls

When repeatedly generating random values (with the same `rand`

parameters), it happens for some types that the result of a computation is used for each call. In this case, the computation can be decoupled from actually generating the values. This is the case for example with the default implementation for `AbstractArray`

. Assume that `rand(rng, 1:20)`

has to be called repeatedly in a loop: the way to take advantage of this decoupling is as follows:

```
rng = MersenneTwister()
sp = Random.Sampler(rng, 1:20) # or Random.Sampler(MersenneTwister,1:20)
for x in X
n = rand(rng, sp) # similar to n = rand(rng, 1:20)
# use n
end
```

This mechanism is of course used by the default implementation of random array generation (like in `rand(1:20, 10)`

). In order to implement this decoupling for a custom type, a helper type can be used. Going back to our `Die`

example: `rand(::Die)`

uses random generation from a range, so there is an opportunity for this optimization:

```
import Random: Sampler, rand
struct SamplerDie <: Sampler{Int} # generates values of type Int
die::Die
sp::Sampler{Int} # this is an abstract type, so this could be improved
end
Sampler(RNG::Type{<:AbstractRNG}, die::Die, r::Random.Repetition) =
SamplerDie(die, Sampler(RNG, 1:die.nsides, r))
# the `r` parameter will be explained later on
rand(rng::AbstractRNG, sp::SamplerDie) = rand(rng, sp.sp)
```

It's now possible to get a sampler with `sp = Sampler(rng, die)`

, and use `sp`

instead of `die`

in any `rand`

call involving `rng`

. In the simplistic example above, `die`

doesn't need to be stored in `SamplerDie`

but this is often the case in practice.

This pattern is so frequent that a helper type named `Random.SamplerSimple`

is available, saving us the definition of `SamplerDie`

: we could have implemented our decoupling with:

```
Sampler(RNG::Type{<:AbstractRNG}, die::Die, r::Random.Repetition) =
SamplerSimple(die, Sampler(RNG, 1:die.nsides, r))
rand(rng::AbstractRNG, sp::SamplerSimple{Die}) = rand(rng, sp.data)
```

Here, `sp.data`

refers to the second parameter in the call to the `SamplerSimple`

constructor (in this case equal to `Sampler(rng, 1:die.nsides, r)`

), while the `Die`

object can be accessed via `sp[]`

.

Another helper type is currently available for other cases, `Random.SamplerTag`

, but is considered as internal API, and can break at any time without proper deprecations.

#### Using distinct algorithms for scalar or array generation

In some cases, whether one wants to generate only a handful of values or a large number of values will have an impact on the choice of algorithm. This is handled with the third parameter of the `Sampler`

constructor. Let's assume we defined two helper types for `Die`

, say `SamplerDie1`

which should be used to generate only few random values, and `SamplerDieMany`

for many values. We can use those types as follows:

```
Sampler(RNG::Type{<:AbstractRNG}, die::Die, ::Val{1}) = SamplerDie1(...)
Sampler(RNG::Type{<:AbstractRNG}, die::Die, ::Val{Inf}) = SamplerDieMany(...)
```

Of course, `rand`

must also be defined on those types (i.e. `rand(::AbstractRNG, ::SamplerDie1)`

and `rand(::AbstractRNG, ::SamplerDieMany)`

).

Note: `Sampler(rng, x)`

is simply a shorthand for `Sampler(rng, x, Val(Inf))`

, and `Random.Repetition`

is an alias for `Union{Val{1}, Val{Inf}}`

.

### Creating new generators

The API is not clearly defined yet, but as a rule of thumb:

- any
`rand`

method producing "basic" types (`isbitstype`

integer and floating types in`Base`

) should be defined for this specific RNG, if they are needed; - other documented
`rand`

methods accepting an`AbstractRNG`

should work out of the box, (provided the methods from 1) what are relied on are implemented), but can of course be specialized for this RNG if there is room for optimization.

Concerning 1), a `rand`

method may happen to work automatically, but it's not officially supported and may break without warnings in a subsequent release.

To define a new `rand`

method for an hypothetical `MyRNG`

generator, and a value specification `s`

(e.g. `s == Int`

, or `s == 1:10`

) of type `S==typeof(s)`

or `S==Type{s}`

if `s`

is a type, the same two methods as we saw before must be defined:

`Sampler(::Type{MyRNG}, ::S, ::Repetition)`

, which returns an object of type say`SamplerS`

`rand(rng::MyRNG, sp::SamplerS)`

It can happen that `Sampler(rng::AbstractRNG, ::S, ::Repetition)`

is already defined in the `Random`

module. It would then be possible to skip step 1) in practice (if one wants to specialize generation for this particular RNG type), but the corresponding `SamplerS`

type is considered as internal detail, and may be changed without warning.

#### Specializing array generation

In some cases, for a given RNG type, generating an array of random values can be more efficient with a specialized method than by merely using the decoupling technique explained before. This is for example the case for `MersenneTwister`

, which natively writes random values in an array.

To implement this specialization for `MyRNG`

and for a specification `s`

, producing elements of type `S`

, the following method can be defined: `rand!(rng::MyRNG, a::AbstractArray{S}, ::SamplerS)`

, where `SamplerS`

is the type of the sampler returned by `Sampler(MyRNG, s, Val(Inf))`

. Instead of `AbstractArray`

, it's possible to implement the functionality only for a subtype, e.g. `Array{S}`

. The non-mutating array method of `rand`

will automatically call this specialization internally.