# Random distributions

For maximum flexibility when producing random values, we define the Distribution trait:

#![allow(unused)]
fn main() {
// a producer of data of type T:
pub trait Distribution<T> {
// the key function:
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> T;

// a convenience function defined using sample:
fn sample_iter<R>(self, rng: R) -> DistIter<Self, R, T>
where
Self: Sized,
R: Rng,
{ ... }
}
}

Rand provides implementations of many different distributions; we cover the most common of these here, but for full details refer to the distributions module and the rand_distr crate.

# Uniform distributions

The most obvious type of distribution is the one we already discussed: one without pattern, where each value or range of values is equally likely. This is known as uniform.

Rand actually has several variants of this, representing different ranges:

• Standard requires no parameters and samples values uniformly according to the type. Rng::gen provides a short-cut to this distribution.
• Uniform is parametrised by Uniform::new(low, high) (including low, excluding high) or Uniform::new_inclusive(low, high) (including both), and samples values uniformly within this range. Rng::gen_range is a convenience method defined over Uniform::sample_single, optimised for single-sample usage.
• Alphanumeric is uniform over the char values 0-9A-Za-z.
• Open01 and OpenClosed01 are provide alternate sampling ranges for floating-point types (see below).

## Uniform sampling by type

Lets go over the distributions by type:

• For bool, Standard samples each value with probability 50%.

• For Option<T>, the Standard distribution samples None with probability 50%, otherwise Some(value) is sampled, according to its type.

• For integers (u8 through to u128, usize, and i* variants), Standard samples from all possible values while Uniform samples from the parameterised range.

• For NonZeroU8 and other "non-zero" types, Standard samples uniformly from all non-zero values (rejection method).

• Wrapping<T> integer types are sampled as for the corresponding integer type by the Standard distribution.

• For floats (f32, f64),

• Standard samples from the half-open range [0, 1) with 24 or 53 bits of precision (for f32 and f64 respectively)
• OpenClosed01 samples from the half-open range (0, 1] with 24 or 53 bits of precision
• Open01 samples from the open range (0, 1) with 23 or 52 bits of precision
• Uniform samples from a given range with 23 or 52 bits of precision
• For the char type, the Standard distribution samples from all available Unicode code points, uniformly; many of these values may not be printable (depending on font support). The Alphanumeric samples from only a-z, A-Z and 0-9 uniformly.

• For tuples and arrays, each element is sampled as above, where supported. The Standard and Uniform distributions each support a selection of these types (up to 12-tuples and 32-element arrays). This includes the empty tuple () and array. When using rustc ≥ 1.51, enable the min_const_gen feature to support arrays larger than 32 elements.

• For SIMD types, each element is sampled as above, for Standard and Uniform (for the latter, low and high parameters are also SIMD types, effectively sampling from multiple ranges simultaneously). SIMD support is gated behind a feature flag.

• For enums, you have to implement uniform sampling yourself. For example, you could use the following approach:

#![allow(unused)]
fn main() {
pub enum Food {
Burger,
Pizza,
Kebab,
}

impl Distribution<Food> for Standard {
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> Food {
let index: u8 = rng.gen_range(0..3);
match index {
0 => Food::Burger,
1 => Food::Pizza,
2 => Food::Kebab,
_ => unreachable!(),
}
}
}
}

# Non-uniform distributions

Non-uniform distributions can be divided into two categories, as follows. Some of these discrete and all of the continuous distributions have been moved from the main rand crate to a dedicated rand_distr crate.

## Discrete non-uniform distributions

Discrete distributions sample from boolean or integer types. As above, these can be sampled uniformly, or, as below, via a non-uniform distribution.

Potentially a discrete distribution could sample directly from a set of discrete values such as a slice or an enum. See the section on Sequences regarding Rand's traits for slice and iterator types. Rand does not provide direct sampling from enums, with the exception of Option (see above).

### Booleans

The Bernoulli distribution is a fancy name for generating a boolean with a given a probability p of being true, or defined via a success : failure ratio. Often this is described as a trial with probability p of success (true).

The methods Rng::gen_bool and Rng::gen_ratio are short-cuts to this distribution.

### Integers

The Binomial distribution is related to the Bernoulli in that it models running n independent trials each with probability p of success, then counts the number of successes.

Note that for large n the Binomial distribution's implementation is much faster than sampling n trials individually.

The Poisson distribution expresses the expected number of events occurring within a fixed interval, given that events occur with fixed rate λ. Poisson distribution sampling generates Float values because Floats are used in the sampling calculations, and we prefer to defer to the user on integer types and the potentially lossy and panicking associated conversions. For example, u64 values can be attained with rng.sample(Poisson) as u64.

Note that out of range float to int conversions with as result in undefined behavior for Rust <1.45 and a saturating conversion for Rust >=1.45.

### Weighted sequences

The WeightedIndex distribution samples an index from sequence of weights. See the Sequences section for convenience wrappers directly sampling a slice element.

For example, weighted sampling could be used to model the colour of a marble sampled from a bucket containing 5 green, 15 red and 80 blue marbles.

Currently the Rand lib only implements sampling with replacement, i.e. repeated sampling assumes the same distribution (that any sampled marble has been replaced). An alternative distribution implementing sampling without replacement has been requested.

Note also that two implementations of WeightedIndex are available; the first is optimised for a small number of samples while alias_method::WeightedIndex is optimised for a large number of samples (where "large" may mean "> 1000"; benchmarks recommended).

## Continuous non-uniform distributions

Continuous distributions model samples drawn from the real number line ℝ, or in some cases a point from a higher dimension (ℝ², ℝ³, etc.). We provide implementations for f64 and for f32 output in most cases, although currently the f32 implementations simply reduce the precision of an f64 sample.

The exponential distribution, Exp, simulates time until decay, assuming a fixed rate of decay (i.e. exponential decay).

The Normal distribution (also known as Gaussian) simulates sampling from the Normal distribution ("Bell curve") with the given mean and standard deviation. The LogNormal is related: for sample X from the log-normal distribution, log(X) is normally distributed; this "skews" the normal distribution to avoid negative values and to have a long positive tail.

The UnitCircle and UnitSphereSurface distributions simulate uniform sampling from the edge of a circle or surface of a sphere.

The Cauchy distribution (also known as the Lorentz distribution) is the distribution of the x-intercept of a ray from point (x0, γ) with uniformly distributed angle.

The Beta distribution is a two-parameter probability distribution, whose output values lie between 0 and 1. The Dirichlet distribution is a generalisation to any positive number of parameters.