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## Reference

### Concepts

#### Introduction

Random numbers are required in a number of different problem domains, such as

- numerics (simulation, Monte-Carlo integration)
- games (non-deterministic enemy behavior)
- security (key generation)
- testing (random coverage in white-box tests)

The Boost Random Number Generator Library provides a framework for random number generators with well-defined properties so that the generators can be used in the demanding numerics and security domains. For a general introduction to random numbers in numerics, see

> "Numerical Recipes in C: The art of scientific computing", William H. Press, Saul A. Teukolsky, William A. Vetterling, Brian P. Flannery, 2nd ed., 1992, pp. 274-328

Depending on the requirements of the problem domain, different variations of random number generators are appropriate:

- non-deterministic random number generator
- pseudo-random number generator
- quasi-random number generator

All variations have some properties in common, these concepts (in the STL sense) are called NumberGenerator and UniformRandomNumberGenerator. Each concept will be defined in a subsequent section.

The goals for this library are the following:

- allow easy integration of third-party random-number generators
- define a validation interface for the generators
- provide easy-to-use front-end classes which model popular distributions
- provide maximum efficiency
- allow control on quantization effects in front-end processing (not yet done)

#### Number Generator

A number generator is a function object (std:20.3 [lib.function.objects]) that takes zero arguments. Each call to `operator()` returns a number. In the following table, X denotes a number generator class returning objects of type T, and u is a value of X.

Table 16.1. NumberGenerator requirements

expression | return type | pre/post-condition
--- | --- | ---
`X::result_type` | `T` | `std::numeric_limits<T>::is_specialized` is `true`, `T` is LessThanComparable
`u.operator()()` | `T` | -

| Note
The NumberGenerator requirements do not impose any restrictions on the characteristics of the returned numbers.

#### Uniform Random Number Generator

A uniform random number generator is a NumberGenerator that provides a sequence of random numbers uniformly distributed on a given range. The range can be compile-time fixed or available (only) after run-time construction of the object.

The tight lower bound of some (finite) set S is the (unique) member l in S, so that for all v in S, l <= v holds. Likewise, the tight upper bound of some (finite) set S is the (unique) member u in S, so that for all v in S, v <= u holds.

In the following table, X denotes a number generator class returning objects of type T, and v is a const value of X.

Table 16.2. UniformRandomNumberGenerator requirements

expression | return type | pre/post-condition
--- | --- | ---
`X::has_fixed_range` | `bool` | compile-time constant; if `true`, the range on which the random numbers are uniformly distributed is known at compile-time and members `min_value` and `max_value` exist. Note: This flag may also be `false` due to compiler limitations
`X::min_value` | `T` | compile-time constant; `min_value` is only defined if `has_fixed_range` is `true`. If it exists, it is equal to `v.min()`.
`X::max_value` | `T` | compile-time constant; `max_value` is only defined if `has_fixed_range` is `true`. If it exists, it is equal to `v.max()`
`v.min()` | `T` | tight lower bound on the set of all values returned by `operator()`. The return value of this function shall not change during the lifetime of the object.
`v.max()` | `T` | if `std::numeric_limits<T>::is_integer`, tight upper bound on the set of all values returned by `operator()`, otherwise, the smallest representable number larger than the tight upper bound on the set of all values returned by `operator()`. In any case, the return value of this function shall not change during the lifetime of the object.

The member functions `min`, `max`, and `operator()` shall have amortized constant time complexity.

| Note
For integer generators (i.e. integer `T`), the generated values `x` fulfill `min() <= x <= max()`, for non-integer generators (i.e. non-integer `T`), the generated values `x` fulfill `min() <= x < max()`. Rationale: The range description with min and max serves two purposes. First, it allows scaling of the values to some canonical range, such as [0..1). Second, it describes the significant bits of the values, which may be relevant for further processing. The range is a closed interval [min,max] for integers, because the underlying type may not be able to represent the half-open interval [min,max+1). It is a half-open interval [min, max) for non-integers, because this is much more practical for borderline cases of continuous distributions.

| Note
The UniformRandomNumberGenerator concept does not require `operator()(long)` and thus it does not fulfill the `RandomNumberGenerator` (std:25.2.11 [lib.alg.random.shuffle]) requirements. Use the `random_number_generator` adapter for that. Rationale: `operator()(long)` is not provided, because mapping the output of some generator with integer range to a different integer range is not trivial.

#### Non-deterministic Uniform Random Number Generator

A non-deterministic uniform random number generator is a UniformRandomNumberGenerator that is based on some stochastic process. Thus, it provides a sequence of truly-random numbers. Examples for such processes are nuclear decay, noise of a Zehner diode, tunneling of quantum particles, rolling a die, drawing from an urn, and tossing a coin. Depending on the environment, inter-arrival times of network packets or keyboard events may be close approximations of stochastic processes.

The class `random_device` is a model for a non-deterministic random number generator.

| Note
This type of random-number generator is useful for security applications, where it is important to prevent an outside attacker from guessing the numbers and thus obtaining your encryption or authentication key. Thus, models of this concept should be cautious not to leak any information, to the extent possible by the environment. For example, it might be advisable to explicitly clear any temporary storage as soon as it is no longer needed.

#### Pseudo-Random Number Generator

A pseudo-random number generator is a UniformRandomNumberGenerator which provides a deterministic sequence of pseudo-random numbers, based on some algorithm and internal state. `Linear congruential` and `inversive congruential` generators are examples of such pseudo-random number generators. Often, these generators are very sensitive to their parameters. In order to prevent wrong implementations from being used, an external testsuite should check that the generated sequence and the validation value provided do indeed match.

Donald E. Knuth gives an extensive overview on pseudo-random number generation in his book "The Art of Computer Programming, Vol. 2, 3rd edition, Addison-Wesley, 1997". The descriptions for the specific generators contain additional references.

| Note
Because the state of a pseudo-random number generator is necessarily finite, the sequence of numbers returned by the generator will loop eventually.

In addition to the UniformRandomNumberGenerator requirements, a pseudo-random number generator has some additional requirements. In the following table, `X` denotes a pseudo-random number generator class returning objects of type `T`, `x` is a value of `T`, `u` is a value of `X`, and `v` is a const value of `X`.

Table 16.3. PseudoRandomNumberGenerator requirements

expression | return type | pre/post-condition
--- | --- | ---
`X()` | - | creates a generator in some implementation-defined state. Note: Several generators thusly created may possibly produce dependent or identical sequences of random numbers.
`explicit X(...)` | - | creates a generator with user-provided state; the implementation shall specify the constructor argument(s)
`u.seed(...)` | `void` | sets the current state according to the argument(s); at least functions with the same signature as the non-default constructor(s) shall be provided.
`X::validation(x)` | `bool` | compares the pre-computed and hardcoded 10001th element in the generator's random number sequence with x. The generator must have been constructed by its default constructor and seed must not have been called for the validation to be meaningful.

| Note
The seed member function is similar to the assign member function in STL containers. However, the naming did not seem appropriate.

Classes which model a pseudo-random number generator shall also model EqualityComparable, i.e. implement `operator==`. Two pseudo-random number generators are defined to be equivalent if they both return an identical sequence of numbers starting from a given state.

Classes which model a pseudo-random number generator should also model the Streamable concept, i.e. implement `operator<<` and `operator>>`. If so, `operator<<` writes all current state of the pseudo-random number generator to the given `ostream` so that `operator>>` can restore the state at a later time. The state shall be written in a platform-independent manner, but it is assumed that the `locales` used for writing and reading be the same. The pseudo-random number generator with the restored state and the original at the just-written state shall be equivalent.

Classes which model a pseudo-random number generator may also model the CopyConstructible and Assignable concepts. However, note that the sequences of the original and the copy are strongly correlated (in fact, they are identical), which may make them unsuitable for some problem domains. Thus, copying pseudo-random number generators is discouraged; they should always be passed by (non-const) reference.

The classes `rand48`, `minstd_rand`, and `mt19937` are models for a pseudo-random number generator.

| Note
This type of random-number generator is useful for numerics, games and testing. The non-zero arguments constructor(s) and the `seed()` member function(s) allow for a user-provided state to be installed in the generator. This is useful for debugging Monte-Carlo algorithms and analyzing particular test scenarios. The Streamable concept allows to save/restore the state of the generator, for example to re-run a test suite at a later time.

#### Random Distribution

A random distribution produces random numbers distributed according to some distribution, given uniformly distributed random values as input. In the following table, `X` denotes a random distribution class returning objects of type `T`, `u` is a value of `X`, `x` is a (possibly const) value of `X`, and `e` is an lvalue of an arbitrary type that meets the requirements of a UniformRandomNumberGenerator, returning values of type `U`.

Table 16.4. Random distribution requirements (in addition to NumberGenerator, CopyConstructible, and Assignable)

expression | return type | pre/post-condition | complexity
--- | --- | --- | ---
`X::input_type` | `U` | - | compile-time
`u.reset()` | `void` | subsequent uses of `u` do not depend on values produced by `e` prior to invoking `reset`. | constant
`u(e)` | `T` | the sequence of numbers returned by successive invocations with the same object `e` is randomly distributed with some probability density function `p(x)` | amortized constant number of invocations of `e`
`os << x` | `std::ostream&` | writes a textual representation for the parameters and additional internal data of the distribution `x` to `os`. post: The `os.fmtflags` and fill character are unchanged. | O(size of state)
`is >> u` | `std::istream&` | restores the parameters and additional internal data of the distribution `u`. pre: `is` provides a textual representation that was previously written by `operator<<` post: The `is.fmtflags` are unchanged. | O(size of state)

Additional requirements: The sequence of numbers produced by repeated invocations of `x(e)` does not change whether or not `os << x` is invoked between any of the invocations `x(e)`. If a textual representation is written using `os << x` and that representation is restored into the same or a different object `y` of the same type using `is >> y`, repeated invocations of `y(e)` produce the same sequence of random numbers as would repeated invocations of `x(e)`.

### Generators

This library provides several pseudo-random number generators. The quality of a pseudo random number generator crucially depends on both the algorithm and its parameters. This library implements the algorithms as class templates with template value parameters, hidden in `namespace boost::random`. Any particular choice of parameters is represented as the appropriately specializing `typedef` in `namespace boost`.

Pseudo-random number generators should not be constructed (initialized) frequently during program execution, for two reasons. First, initialization requires full initialization of the internal state of the generator. Thus, generators with a lot of internal state (see below) are costly to initialize. Second, initialization always requires some value used as a "seed" for the generated sequence. It is usually difficult to obtain several good seed values. For example, one method to obtain a seed is to determine the current time at the highest resolution available, e.g. microseconds or nanoseconds. When the pseudo-random number generator is initialized again with the then-current time as the seed, it is likely that this is at a near-constant (non-random) distance from the time given as the seed for first initialization. The distance could even be zero if the resolution of the clock is low, thus the generator re-iterates the same sequence of random numbers. For some applications, this is inappropriate.

Note that all pseudo-random number generators described below are CopyConstructible and Assignable. Copying or assigning a generator will copy all its internal state, so the original and the copy will generate the identical sequence of random numbers. Often, such behavior is not wanted. In particular, beware of the algorithms from the standard library such as `std::generate`. They take a functor argument by value, thereby invoking the copy constructor when called.

The following table gives an overview of some characteristics of the generators. The cycle length is a rough estimate of the quality of the generator; the approximate relative speed is a performance measure, higher numbers mean faster random number generation.

Table 16.5. generators

generator | length of cycle | approx. memory requirements | approx. speed compared to fastest | comment
--- | --- | --- | --- | ---
`minstd_rand0` | 231-2 | `sizeof(int32_t)` | 30% | -
`minstd_rand` | 231-2 | `sizeof(int32_t)` | 29% | -
`rand48` | 248-1 | `sizeof(uint64_t)` | 100% | -
`ecuyer1988` | approx. 261 | `2*sizeof(int32_t)` | 19% | -
`kreutzer1986` | ? | `1368*sizeof(uint32_t)` | 31% | -
`taus88` | ~288 | `3*sizeof(uint32_t)` | 78% | -
`hellekalek1995` | 231-1 | `sizeof(int32_t)` | 0% | good uniform distribution in several dimensions
`mt11213b` | 211213-1 | `352*sizeof(uint32_t)` | 44% | good uniform distribution in up to 350 dimensions
`mt19937` | 219937-1 | `625*sizeof(uint32_t)` | 44% | good uniform distribution in up to 623 dimensions
`lagged_fibonacci607` | ~232000 | `607*sizeof(double)` | 26% | -
`lagged_fibonacci1279` | ~267000 | `1279*sizeof(double)` | 26% | -
`lagged_fibonacci2281` | ~2120000 | `2281*sizeof(double)` | 26% | -
`lagged_fibonacci3217` | ~2170000 | `3217*sizeof(double)` | 26% | -
`lagged_fibonacci4423` | ~2230000 | `4423*sizeof(double)` | 25% | -
`lagged_fibonacci9689` | ~2510000 | `9689*sizeof(double)` | 25% | -
`lagged_fibonacci19937` | ~21050000 | `19937*sizeof(double)` | 25% | -
`lagged_fibonacci23209` | ~21200000 | `23209*sizeof(double)` | 25% | -
`lagged_fibonacci44497` | ~22300000 | `44497*sizeof(double)` | 25% | -
`ranlux3` | ~10171 | `24*sizeof(int)` | 2% | -
`ranlux4` | ~10171 | `24*sizeof(int)` | 1% | -
`ranlux64_3` | ~10171 | `24*sizeof(int64_t)` | 2% | -
`ranlux64_4` | ~10171 | `24*sizeof(int64_t)` | 1% | -
`ranlux3_01` | ~10171 | `24*sizeof(float)` | 2% | -
`ranlux4_01` | ~10171 | `24*sizeof(float)` | 1% | -
`ranlux64_3_01` | ~10171 | `24*sizeof(double)` | 2% | -
`ranlux64_4_01` | ~10171 | `24*sizeof(double)` | 1% | -

As observable from the table, there is generally a quality/performance/memory trade-off to be decided upon when choosing a random-number generator. The multitude of generators provided in this library allows the application programmer to optimize the trade-off with regard to his application domain. Additionally, employing several fundamentally different random number generators for a given application of Monte Carlo simulation will improve the confidence in the results.

If the names of the generators don't ring any bell and you have no idea which generator to use, it is reasonable to employ `mt19937` for a start: It is fast and has acceptable quality.

| Note
These random number generators are not intended for use in applications where non-deterministic random numbers are required. See `random_device` for a choice of (hopefully) non-deterministic random number generators.

### Distributions

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.

Table 16.6. distributions

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 radians (assuming the angle is uniformly distributed)
`bernoulli_distribution` | Bernoulli experiment: discrete boolean valued distribution with configurable probability | tossing a coin (p=0.5)
`binomial_distribution` | counts outcomes of repeated Bernoulli experiments | tossing a coin 20 times and counting how many front sides are shown
`cauchy_distribution` | cauchy distribution | -
`gamma_distribution` | gamma distribution | -
`poisson_distribution` | poisson distribution | counting the number of alpha particles emitted by radioactive matter in a fixed period of time
`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` | 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

### Headers

#### Header <boost/nondet_random.hpp>

    namespace boost {
      class random_device;
    }

#### Header <boost/random.hpp>

    namespace boost {
      typedef random::xor_combine< random::xor_combine< random::linear_feedback_shift< uint32_t, 32, 31, 13, 12, 0 >, 0, random::linear_feedback_shift< uint32_t, 32, 29, 2, 4, 0 >, 0, 0 >, 0, random::linear_feedback_shift< uint32_t, 32, 28, 3, 17, 0 >, 0, 0 > taus88;
    }

#### Header <boost/random/additive_combine.hpp>

    namespace boost {
      typedef random::additive_combine< random::linear_congruential< int32_t, 40014, 0, 2147483563, 0 >, random::linear_congruential< int32_t, 40692, 0, 2147483399, 0 >, 2060321752 > ecuyer1988;
      namespace random {
        template<typename MLCG1, typename MLCG2, typename MLCG1::result_type val>
          class additive_combine;
      }
    }

#### Header <boost/random/bernoulli_distribution.hpp>

    namespace boost {
      template<typename RealType = double> class bernoulli_distribution;
    }

#### Header <boost/random/binomial_distribution.hpp>

    namespace boost {
      template<typename IntType = int, typename RealType = double>
        class binomial_distribution;
    }

#### Header <boost/random/cauchy_distribution.hpp>

    namespace boost {
      template<typename RealType = double> class cauchy_distribution;
    }

#### Header <boost/random/discard_block.hpp>

    namespace boost {
      namespace random {
        template<typename UniformRandomNumberGenerator, unsigned int p,
                 unsigned int r>
          class discard_block;
      }
    }

#### Header <boost/random/exponential_distribution.hpp>

    namespace boost {
      template<typename RealType = double> class exponential_distribution;
    }

#### Header <boost/random/gamma_distribution.hpp>

    namespace boost {
      template<typename RealType = double> class gamma_distribution;
    }

#### Header <boost/random/geometric_distribution.hpp>

    namespace boost {
      template<typename IntType = int, typename RealType = double>
        class geometric_distribution;
    }

#### Header <boost/random/inversive_congruential.hpp>

    namespace boost {
      typedef random::inversive_congruential< int32_t, 9102, 2147483647-36884165, 2147483647, 0 > hellekalek1995;
      namespace random {
        template<typename IntType, IntType a, IntType b, IntType p, IntType val>
          class inversive_congruential;
      }
    }

#### Header <boost/random/lagged_fibonacci.hpp>

    namespace boost {
      typedef random::lagged_fibonacci_01< double, 48, 607, 273 > lagged_fibonacci607;
      typedef random::lagged_fibonacci_01< double, 48, 1279, 418 > lagged_fibonacci1279;
      typedef random::lagged_fibonacci_01< double, 48, 2281, 1252 > lagged_fibonacci2281;
      typedef random::lagged_fibonacci_01< double, 48, 3217, 576 > lagged_fibonacci3217;
      typedef random::lagged_fibonacci_01< double, 48, 4423, 2098 > lagged_fibonacci4423;
      typedef random::lagged_fibonacci_01< double, 48, 9689, 5502 > lagged_fibonacci9689;
      typedef random::lagged_fibonacci_01< double, 48, 19937, 9842 > lagged_fibonacci19937;
      typedef random::lagged_fibonacci_01< double, 48, 23209, 13470 > lagged_fibonacci23209;
      typedef random::lagged_fibonacci_01< double, 48, 44497, 21034 > lagged_fibonacci44497;
      namespace random {
        template<typename UIntType, int w, unsigned int p, unsigned int q,
                 UIntType val = 0>
          class lagged_fibonacci;
        template<typename RealType, int w, unsigned int p, unsigned int q>
          class lagged_fibonacci_01;
      }
    }

#### Header <boost/random/linear_congruential.hpp>

    namespace boost {
      class rand48;
      typedef random::linear_congruential< int32_t, 16807, 0, 2147483647, 1043618065 > minstd_rand0;
      typedef random::linear_congruential< int32_t, 48271, 0, 2147483647, 399268537 > minstd_rand;
      namespace random {
        template<typename IntType, IntType a, IntType c, IntType m, IntType val>
          class linear_congruential;
      }
    }

#### Header <boost/random/linear_feedback_shift.hpp>

    namespace boost {
      namespace random {
        template<typename UIntType, int w, int k, int q, int s, UIntType val>
          class linear_feedback_shift;
      }
    }

#### Header <boost/random/lognormal_distribution.hpp>

    namespace boost {
      template<typename RealType = double> class lognormal_distribution;
    }

#### Header <boost/random/mersenne_twister.hpp>

    namespace boost {
      typedef random::mersenne_twister< uint32_t, 32, 351, 175, 19, 0xccab8ee7, 11, 7, 0x31b6ab00, 15, 0xffe50000, 17, 0xa37d3c92 > mt11213b;
      typedef random::mersenne_twister< uint32_t, 32, 624, 397, 31, 0x9908b0df, 11, 7, 0x9d2c5680, 15, 0xefc60000, 18, 3346425566U > mt19937;
      namespace random {
        template<typename UIntType, int w, int n, int m, int r, UIntType a, int u,
                 int s, UIntType b, int t, UIntType c, int l, UIntType val>
          class mersenne_twister;
      }
    }

#### Header <boost/random/normal_distribution.hpp>

    namespace boost {
      template<typename RealType = double> class normal_distribution;
    }

#### Header <boost/random/poisson_distribution.hpp>

    namespace boost {
      template<typename IntType = int, typename RealType = double>
        class poisson_distribution;
    }

#### Header <boost/random/random_number_generator.hpp>

    namespace boost {
      template<typename UniformRandomNumberGenerator, typename IntType = long>
        class random_number_generator;
    }

#### Header <boost/random/ranlux.hpp>

    namespace boost {
      typedef random::discard_block< random::ranlux_base, 223, 24 > ranlux3;
      typedef random::discard_block< random::ranlux_base, 389, 24 > ranlux4;
      typedef random::discard_block< random::ranlux_base_01, 223, 24 > ranlux3_01;
      typedef random::discard_block< random::ranlux_base_01, 389, 24 > ranlux4_01;
      typedef random::discard_block< random::ranlux64_base_01, 223, 24 > ranlux64_3_01;
      typedef random::discard_block< random::ranlux64_base_01, 389, 24 > ranlux64_4_01;
      typedef random::discard_block< random::ranlux64_base, 223, 24 > ranlux64_3;
      typedef random::discard_block< random::ranlux64_base, 389, 24 > ranlux64_4;
      namespace random {
        typedef subtract_with_carry< int,(1<< 24), 10, 24, 0 > ranlux_base;
        typedef subtract_with_carry_01< float, 24, 10, 24 > ranlux_base_01;
        typedef subtract_with_carry_01< double, 48, 10, 24 > ranlux64_base_01;
        typedef random::subtract_with_carry< int64_t,(int64_t(1)<< 48), 10, 24, 0 > ranlux64_base;
      }
    }

#### Header <boost/random/shuffle_output.hpp>

    namespace boost {
      typedef random::shuffle_output< random::linear_congruential< uint32_t, 1366, 150889, 714025, 0 >, 97, 139726 > kreutzer1986;
      namespace random {
        template<typename UniformRandomNumberGenerator, int k,
                 typename UniformRandomNumberGenerator::result_type val = 0>
          class shuffle_output;
      }
    }

#### Header <boost/random/subtract_with_carry.hpp>

    namespace boost {
      namespace random {
        template<typename IntType, IntType m, unsigned int s, unsigned int r,
                 IntType val>
          class subtract_with_carry;
        template<typename RealType, int w, unsigned int s, unsigned int r,
                 int val = 0>
          class subtract_with_carry_01;
      }
    }

#### Header <boost/random/triangle_distribution.hpp>

    namespace boost {
      template<typename RealType = double> class triangle_distribution;
    }

#### Header <boost/random/uniform_01.hpp>

    namespace boost {
      template<typename RealType = double> class uniform_01;
    }

#### Header <boost/random/uniform_int.hpp>

    namespace boost {
      template<typename IntType = int> class uniform_int;
    }

#### Header <boost/random/uniform_on_sphere.hpp>

    namespace boost {
      template<typename RealType = double, typename Cont = std::vector<RealType> >
        class uniform_on_sphere;
    }

#### Header <boost/random/uniform_real.hpp>

    namespace boost {
      template<typename RealType = double> class uniform_real;
    }

#### Header <boost/random/uniform_smallint.hpp>

    namespace boost {
      template<typename IntType = int> class uniform_smallint;
    }

#### Header <boost/random/variate_generator.hpp>

    namespace boost {
      template<typename Engine, typename Distribution> class variate_generator;
    }

#### Header <boost/random/xor_combine.hpp>

    namespace boost {
      namespace random {
        template<typename URNG1, int s1, typename URNG2, int s2> class xor_combine;
      }
    }