# Random Number Library Distributions

## Introduction

In addition to the random number generators, this library provides distribution functions which map one distribution (often a uniform distribution provided by some generator) to another.

Usually, there are several possible implementations of any given mapping. Often, there is a choice between using more space, more invocations of the underlying source of random numbers, or more time-consuming arithmetic such as trigonometric functions. This interface description does not mandate any specific implementation. However, implementations which cannot reach certain values of the specified distribution or otherwise do not converge statistically to it are not acceptable.

distribution explanation example
`uniform_smallint` discrete uniform distribution on a small set of integers (much smaller than the range of the underlying generator) drawing from an urn
`uniform_int` discrete uniform distribution on a set of integers; the underlying generator may be called several times to gather enough randomness for the output drawing from an urn
`uniform_01` continuous uniform distribution on the range [0,1); important basis for other distributions -
`uniform_real` continuous uniform distribution on some range [min, max) of real numbers for the range [0, 2pi): randomly dropping a stick and measuring its angle in radiants (assuming the angle is uniformly distributed)
`bernoulli_distribution` Bernoulli experiment: discrete boolean valued distribution with configurable probability tossing a coin (p=0.5)
`geometric_distribution` measures distance between outcomes of repeated Bernoulli experiments throwing a die several times and counting the number of tries until a "6" appears for the first time
`triangle_distribution` ? ?
`exponential_distribution` exponential distribution measuring the inter-arrival time of alpha particles emitted by radioactive matter
`normal_distribution` counts outcomes of (infinitely) repeated Bernoulli experiments tossing a coin 10000 times and counting how many front sides are shown
`lognormal_distribution` lognormal distribution (sometimes used in simulations) measuring the job completion time of an assembly line worker
`uniform_on_sphere` uniform distribution on a unit sphere of arbitrary dimension choosing a random point on Earth (assumed to be a sphere) where to spend the next vacations

The template parameters of the distribution functions are always in the order

• Underlying source of random numbers
• If applicable, return type, with a default to a reasonable type.

The distribution functions no longer satisfy the input iterator requirements (std:24.1.1 [lib.input.iterators]), because this is redundant given the Generator interface and imposes a run-time overhead on all users. Moreover, a Generator interface appeals to random number generation as being more "natural". Use an iterator adaptor if you need to wrap any of the generators in an input iterator interface.

All of the distribution functions described below store a non-const reference to the underlying source of random numbers. Therefore, the distribution functions are not Assignable. However, they are CopyConstructible. Copying a distribution function will copy the parameter values. Furthermore, both the copy and the original will refer to the same underlying source of random numbers. Therefore, both the copy and the original will obtain their underlying random numbers from a single sequence.

In this description, I have refrained from documenting those members in detail which are already defined in the concept documentation.

## Synopsis of the distributions available from header `<boost/random.hpp>`

```namespace boost {
template<class IntType = int>
class uniform_smallint;
template<class IntType = int>
class uniform_int;
template<class RealType = double>
class uniform_01;
template<class RealType = double>
class uniform_real;

// discrete distributions
template<class RealType = double>
class bernoulli_distribution;
template<class IntType = int>
class geometric_distribution;

// continuous distributions
template<class RealType = double>
class triangle_distribution;
template<class RealType = double>
class exponential_distribution;
template<class RealType = double>
class normal_distribution;
template<class RealType = double>
class lognormal_distribution;
template<class RealType = double,
class Cont = std::vector<RealType> >
class uniform_on_sphere;
}
```

## Class template `uniform_smallint`

### Synopsis

```#include <boost/random/uniform_smallint.hpp>

template<class IntType = int>
class uniform_smallint
{
public:
typedef IntType input_type;
typedef IntType result_type;
static const bool has_fixed_range = false;
uniform_smallint(IntType min, IntType max);
result_type min() const;
result_type max() const;
void reset();
template<class UniformRandomNumberGenerator>
result_type operator()(UniformRandomNumberGenerator& urng);
};
```

### Description

The distribution function `uniform_smallint` models a random distribution. On each invocation, it returns a random integer value uniformly distributed in the set of integer numbers {min, min+1, min+2, ..., max}. It assumes that the desired range (max-min+1) is small compared to the range of the underlying source of random numbers and thus makes no attempt to limit quantization errors.

Let rout=(max-min+1) the desired range of integer numbers, and let rbase be the range of the underlying source of random numbers. Then, for the uniform distribution, the theoretical probability for any number i in the range rout will be pout(i) = 1/rout. Likewise, assume a uniform distribution on rbase for the underlying source of random numbers, i.e. pbase(i) = 1/rbase. Let pout_s(i) denote the random distribution generated by `uniform_smallint`. Then the sum over all i in rout of (pout_s(i)/pout(i) -1)2 shall not exceed rout/rbase2 (rbase mod rout)(rout - rbase mod rout).

The template parameter `IntType` shall denote an integer-like value type.

Note: The property above is the square sum of the relative differences in probabilities between the desired uniform distribution pout(i) and the generated distribution pout_s(i). The property can be fulfilled with the calculation (base_rng mod rout), as follows: Let r = rbase mod rout. The base distribution on rbase is folded onto the range rout. The numbers i < r have assigned (rbase div rout)+1 numbers of the base distribution, the rest has only (rbase div rout). Therefore, pout_s(i) = ((rbase div rout)+1) / rbase for i < r and pout_s(i) = (rbase div rout)/rbase otherwise. Substituting this in the above sum formula leads to the desired result.

Note: The upper bound for (rbase mod rout)(rout - rbase mod rout) is rout2/4. Regarding the upper bound for the square sum of the relative quantization error of rout3/(4*rbase2), it seems wise to either choose rbase so that rbase > 10*rout2 or ensure that rbase is divisible by rout.

### Members

```uniform_smallint(IntType min, IntType max)
```

Effects: Constructs a `uniform_smallint` functor. `min` and `max` are the lower and upper bounds of the output range, respectively.

## Class template `uniform_int`

### Synopsis

```#include <boost/random/uniform_int.hpp>

template<class IntType = int>
class uniform_int
{
public:
typedef IntType input_type;
typedef IntType result_type;
static const bool has_fixed_range = false;
explicit uniform_int(IntType min = 0, IntType max = 9);
result_type min() const;
result_type max() const;
void reset();
template<class UniformRandomNumberGenerator>
result_type operator()(UniformRandomNumberGenerator& urng);
template<class UniformRandomNumberGenerator>
result_type operator()(UniformRandomNumberGenerator& urng, result_type n);
};
```

### Description

The distribution function `uniform_int` models a random distribution. On each invocation, it returns a random integer value uniformly distributed in the set of integer numbers {min, min+1, min+2, ..., max}.

The template parameter `IntType` shall denote an integer-like value type.

### Members

```    uniform_int(IntType min = 0, IntType max = 9)
```

Requires: min <= max
Effects: Constructs a `uniform_int` object. `min` and `max` are the parameters of the distribution.

```    result_type min() const
```

Returns: The "min" parameter of the distribution.

```    result_type max() const
```

Returns: The "max" parameter of the distribution.

```    result_type operator()(UniformRandomNumberGenerator& urng, result_type
n)
```

Returns: A uniform random number x in the range 0 <= x < n. [Note: This allows a `variate_generator` object with a `uniform_int` distribution to be used with std::random_shuffe, see [lib.alg.random.shuffle]. ]

## Class template `uniform_01`

### Synopsis

```#include <boost/random/uniform_01.hpp>

template<class UniformRandomNumberGenerator, class RealType = double>
class uniform_01
{
public:
typedef UniformRandomNumberGenerator base_type;
typedef RealType result_type;
static const bool has_fixed_range = false;
explicit uniform_01(base_type rng);
result_type operator()();
result_type min() const;
result_type max() const;
};
```

### Description

The distribution function `uniform_01` models a random distribution. On each invocation, it returns a random floating-point value uniformly distributed in the range [0..1). The value is computed using `std::numeric_limits<RealType>::digits` random binary digits, i.e. the mantissa of the floating-point value is completely filled with random bits. [Note: Should this be configurable?]

WARNING: As an exception / historic accident, this class takes a UniformRandomNumberGenerator as its constructor parameter, and BY VALUE. Usually, you want reference semantics so that the state of the passed-in generator is changed in-place and not copied. In that case, explicitly supply a reference type for the template parameter UniformRandomNumberGenerator.

The template parameter `RealType` shall denote a float-like value type with support for binary operators +, -, and /. It must be large enough to hold floating-point numbers of value `rng.max()-rng.min()+1`.

`base_type::result_type` must be a number-like value type, it must support `static_cast<>` to `RealType` and binary operator -.

Note: The current implementation is buggy, because it may not fill all of the mantissa with random bits. I'm unsure how to fill a (to-be-invented) `boost::bigfloat` class with random bits efficiently. It's probably time for a traits class.

### Members

```explicit uniform_01(base_type rng)
```

Effects: Constructs a `uniform_01` functor with the given uniform random number generator as the underlying source of random numbers.

## Class template `uniform_real`

### Synopsis

```#include <boost/random/uniform_real.hpp>

template<class RealType = double>
class uniform_real
{
public:
typedef RealType input_type;
typedef RealType result_type;
static const bool has_fixed_range = false;
uniform_real(RealType min = RealType(0), RealType max = RealType(1));
result_type min() const;
result_type max() const;
void reset();
template<class UniformRandomNumberGenerator>
result_type operator()(UniformRandomNumberGenerator& urng);
};
```

### Description

The distribution function `uniform_real` models a random distribution. On each invocation, it returns a random floating-point value uniformly distributed in the range [min..max). The value is computed using `std::numeric_limits<RealType>::digits` random binary digits, i.e. the mantissa of the floating-point value is completely filled with random bits.

Note: The current implementation is buggy, because it may not fill all of the mantissa with random bits.

### Members

```    uniform_real(RealType min = RealType(0), RealType max = RealType(1))
```

Requires: min <= max
Effects: Constructs a `uniform_real` object; `min` and `max` are the parameters of the distribution.

```    result_type min() const
```

Returns: The "min" parameter of the distribution.

```    result_type max() const
```

Returns: The "max" parameter of the distribution.

## Class template `bernoulli_distribution`

### Synopsis

```#include <boost/random/bernoulli_distribution.hpp>

template<class RealType = double>
class bernoulli_distribution
{
public:
typedef int input_type;
typedef bool result_type;

explicit bernoulli_distribution(const RealType& p = RealType(0.5));
RealType p() const;
void reset();
template<class UniformRandomNumberGenerator>
result_type operator()(UniformRandomNumberGenerator& urng);
};
```

### Description

Instantiations of class template `bernoulli_distribution` model a random distribution. Such a random distribution produces `bool` values distributed with probabilities P(true) = p and P(false) = 1-p. p is the parameter of the distribution.

### Members

```    bernoulli_distribution(const RealType& p = RealType(0.5))
```

Requires: 0 <= p <= 1
Effects: Constructs a `bernoulli_distribution` object. `p` is the parameter of the distribution.

```    RealType p() const
```

Returns: The "p" parameter of the distribution.

## Class template `geometric_distribution`

### Synopsis

```#include <boost/random/geometric_distribution.hpp>

template<class UniformRandomNumberGenerator, class IntType = int>
class geometric_distribution
{
public:
typedef RealType input_type;
typedef IntType result_type;

explicit geometric_distribution(const RealType& p = RealType(0.5));
RealType p() const;
void reset();
template<class UniformRandomNumberGenerator>
result_type operator()(UniformRandomNumberGenerator& urng);
};
```

### Description

Instantiations of class template `geometric_distribution` model a random distribution. A `geometric_distribution` random distribution produces integer values i >= 1 with p(i) = (1-p) * pi-1. p is the parameter of the distribution. Each invocation of the UniformRandomNumberGenerator shall result in a floating-point value in the range [0,1).

### Members

```    geometric_distribution(const RealType& p = RealType(0.5))
```

Requires: 0 < p < 1
Effects: Constructs a `geometric_distribution` object; `p` is the parameter of the distribution.

```   RealType p() const
```

Returns: The "p" parameter of the distribution.

## Class template `triangle_distribution`

### Synopsis

```#include <boost/random/triangle_distribution.hpp>

template<class RealType = double>
class triangle_distribution
{
public:
typedef RealType input_type;
typedef RealType result_type;
triangle_distribution(result_type a, result_type b, result_type c);
result_type a() const;
result_type b() const;
result_type c() const;
void reset();
template<class UniformRandomNumberGenerator>
result_type operator()(UniformRandomNumberGenerator& urng);
};
```

### Description

Instantiations of class template `triangle_distribution` model a random distribution. The returned floating-point values `x` satisfy ```a <= x <= c```; `x` has a triangle distribution, where `b` is the most probable value for `x`. Each invocation of the UniformRandomNumberGenerator shall result in a floating-point value in the range [0,1).

### Members

```triangle_distribution(result_type a, result_type b, result_type c)
```

Effects: Constructs a `triangle_distribution` functor. `a, b, c` are the parameters for the distribution.

## Class template `exponential_distribution`

### Synopsis

```#include <boost/random/exponential_distribution.hpp>

template<class RealType = double>
class exponential_distribution
{
public:
typedef RealType input_type;
typedef RealType result_type;
explicit exponential_distribution(const result_type& lambda);
RealType lambda() const;
void reset();
template<class UniformRandomNumberGenerator>
result_type operator()(UniformRandomNumberGenerator& urng);
};
```

### Description

Instantiations of class template `exponential_distribution` model a random distribution. Such a distribution produces random numbers x > 0 distributed with probability density function p(x) = lambda * exp(-lambda * x), where lambda is the parameter of the distribution. Each invocation of the UniformRandomNumberGenerator shall result in a floating-point value in the range [0,1).

### Members

```    exponential_distribution(const result_type& lambda = result_type(1))
```

Requires: lambda > 0
Effects: Constructs an `exponential_distribution` object with `rng` as the reference to the underlying source of random numbers. `lambda` is the parameter for the distribution.

```    RealType lambda() const
```

Returns: The "lambda" parameter of the distribution.

## Class template `normal_distribution`

### Synopsis

```#include <boost/random/normal_distribution.hpp>

template<class RealType = double>
class normal_distribution
{
public:
typedef RealType input_type;
typedef RealType result_type;
explicit normal_distribution(const result_type& mean = 0,
const result_type& sigma = 1);
RealType mean() const;
RealType sigma() const;
void reset();
template<class UniformRandomNumberGenerator>
result_type operator()(UniformRandomNumberGenerator& urng);
};
```

### Description

Instantiations of class template `normal_distribution` model a random distribution. Such a distribution produces random numbers x distributed with probability density function p(x) = 1/sqrt(2*pi*sigma) * exp(- (x-mean)2 / (2*sigma2) ), where mean and sigma are the parameters of the distribution. Each invocation of the UniformRandomNumberGenerator shall result in a floating-point value in the range [0,1).

### Members

```    explicit normal_distribution(const result_type& mean = 0,
const result_type& sigma = 1);
```

Requires: sigma > 0
Effects: Constructs a `normal_distribution` object; `mean` and `sigma` are the parameters for the distribution.

```    RealType mean() const
```

Returns: The "mean" parameter of the distribution.

```    RealType sigma() const
```

Returns: The "sigma" parameter of the distribution.

## Class template `lognormal_distribution`

### Synopsis

```#include <boost/random/lognormal_distribution.hpp>

template<class RealType = double>
class lognormal_distribution
{
public:
typedef typename normal_distribution<RealType>::input_type
typedef RealType result_type;
explicit lognormal_distribution(const result_type& mean = 1.0,
const result_type& sigma = 1.0);
RealType& mean() const;
RealType& sigma() const;
void reset();
template<class UniformRandomNumberGenerator>
result_type operator()(UniformRandomNumberGenerator& urng);
};
```

### Description

Instantiations of class template `lognormal_distribution` model a random distribution. Such a distribution produces random numbers with p(x) = 1/(x * normal_sigma * sqrt(2*pi)) * exp( -(log(x)-normal_mean)2 / (2*normal_sigma2) ) for x > 0, where normal_mean = log(mean2/sqrt(sigma2 + mean2)) and normal_sigma = sqrt(log(1 + sigma2/mean2)). Each invocation of the UniformRandomNumberGenerator shall result in a floating-point value in the range [0,1).

### Members

```lognormal_distribution(const result_type& mean,
const result_type& sigma)
```

Effects: Constructs a `lognormal_distribution` functor. `mean` and `sigma` are the mean and standard deviation of the lognormal distribution.

## Class template `uniform_on_sphere`

### Synopsis

```#include <boost/random/uniform_on_sphere.hpp>

template<class RealType = double,
class Cont = std::vector<RealType> >
class uniform_on_sphere
{
public:
typedef RealType input_type;
typedef Cont result_type;
explicit uniform_on_sphere(int dim = 2);
void reset();
template<class UniformRandomNumberGenerator>
const result_type & operator()(UniformRandomNumberGenerator& urng);
};
```

### Description

Instantiations of class template `uniform_on_sphere` model a random distribution. Such a distribution produces random numbers uniformly distributed on the unit sphere of arbitrary dimension `dim`. The `Cont` template parameter must be a STL-like container type with `begin` and `end` operations returning non-const ForwardIterators of type `Cont::iterator`. Each invocation of the UniformRandomNumberGenerator shall result in a floating-point value in the range [0,1).

### Members

```explicit uniform_on_sphere(int dim = 2)
```

Effects: Constructs a `uniform_on_sphere` functor. `dim` is the dimension of the sphere.

Revised 05 December, 2006