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.

distributionexplanationexample
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

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?]

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.

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.

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.

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.

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)).

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.

Members

explicit uniform_on_sphere(int dim = 2)
Effects: Constructs a uniform_on_sphere functor. dim is the dimension of the sphere.

Jens Maurer, 2003-10-25