# C++ concepts: RandomNumberDistribution

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A `RandomNumberDistribution` is a function object returning random numbers according to a probability density function p(x) or a discrete probability distribution P(x
i).

### Requirements

The type `D` satisfies `RandomNumberDistribution` if

- `D` satisfies `CopyConstructible`
- `D` satisfies `CopyAssignable`

Given

- `T`, the type named by D::result_type
- `P`, the type named by D::param_type, which

  - satisfies `CopyConstructible`
  - satisfies `CopyAssignable`
  - satisfies `EqualityComparable`
  - has a constructor taking identical arguments as each of the constructors of `D` that take arguments corresponding to the distribution parameters.
  - has a member function with the identical name, type, and semantics, as every member function of `D` that returns a parameter of the distribution
  - declares a member typedef using distribution_type = D;

- `d`, a value of type `D`
- `x` and `y`, (possibly const) values of type `D`
- `p`, a (possibly const) value of type `P`
- `g`, `g1`, `g2`, lvalues of a type satisfying `UniformRandomBitGenerator`
- `os`, lvalue of a specialization of std::basic_ostream
- `is`, lvalue of a specialization of std::basic_istream

The following expressions must be valid and have their specified effects

Expression | Type | Notes | Complexity
--- | --- | --- | ---
`D::result_type` | `T` | An arithmetic type | compile-time
`D::param_type` | `P` |  | compile-time
`D()` |  | creates a distribution indistinguishable from any other default-constructed `D` | constant
`D(p)` |  | creates a distribution indistinguishable from `D` constructed directly from the values used to construct `p` | Same as `p`'s construction
`d.reset()` | `void` | Resets the internal state of the distribution. The next call to `operator()` on `d` will not depend on values produced by any engine prior to `reset()`. | constant
`x.param()` | `P` | Returns `p` such that D(p).param() == p | No worse than `D(p)`
`d.param(p)` | `void` | Postcondition: d.param() == p | No worse than `D(p)`
`d(g)` | `T` | The sequence of numbers returned by successive invocations of this call with the same `g` are randomly distributed according to the distribution parametrized by `d.param()` | Amortized constant number of invocations of `g`
`d(g,p)` | `T` | The sequence of numbers returned by successive invocations of this call with the same `g` are randomly distributed according to the distribution parametrized by `p` | Amortized constant number of invocations of `g`
`x.min()` | `T` | the greatest lower bound on the values potentially returned by `x`’s `operator()`, as determined by the current values of `x`’s parameters | constant
`x.max()` | `T` | the least upper bound on the values potentially returned by `x`’s `operator()`, as determined by the current values of `x`’s parameters | constant
`x == y` | `bool` | Establishes an equivalence relation. Returns `true` if x.param() == y.param() and future infinite sequences of values that would be generated by repeated invocations of x(g1) and y(g2) would be equal as long as g1 == g2. | constant
`x != y` | `bool` | `!(x == y)` | constant
`os << x` | Reference to the type of `os` | Writes a textual representation of the distribution parameters and internal state to `os`. The formatting flags and fill character of `os` are unchanged. |

`is >> d`

Reference to the type of `is`

Restores the distribution parameters and internal state with data read from `is`. The formatting flags of `is` are unchanged. The data must have been written using a stream with the same locale, `CharT` and `Traits` stream template parameters, otherwise the behavior is undefined. If bad input is encountered, is.setstate(std::ios::failbit) is called, which may throw std::ios_base::failure. `d` is unchanged in that case.

### Notes

The parameters of a distribution object may be changed either permanently, by using d.param(p) or just for the duration of a single operator() call, by using d(g,p).

Calls to const member functions of the distribution and os << d do not affect the sequence of numbers produced by repeated d(g).

### Standard library

The following standard library components satisfy `RandomNumberDistribution`

uniform_int_distribution (C++11) | produces integer values evenly distributed across a range (class template)
uniform_real_distribution (C++11) | produces real values evenly distributed across a range (class template)
bernoulli_distribution (C++11) | produces bool values on a Bernoulli distribution. (class)
binomial_distribution (C++11) | produces integer values on a binomial distribution. (class template)
negative_binomial_distribution (C++11) | produces integer values on a negative binomial distribution. (class template)
geometric_distribution (C++11) | produces integer values on a geometric distribution. (class template)
poisson_distribution (C++11) | produces integer values on a poisson distribution. (class template)
exponential_distribution (C++11) | produces real values on an exponential distribution. (class template)
gamma_distribution (C++11) | produces real values on an gamma distribution. (class template)
weibull_distribution (C++11) | produces real values on a Weibull distribution. (class template)
extreme_value_distribution (C++11) | produces real values on an extreme value distribution. (class template)
normal_distribution (C++11) | produces real values on a standard normal (Gaussian) distribution. (class template)
lognormal_distribution (C++11) | produces real values on a lognormal distribution. (class template)
chi_squared_distribution (C++11) | produces real values on a chi-squared distribution. (class template)
cauchy_distribution (C++11) | produces real values on a Cauchy distribution. (class template)
fisher_f_distribution (C++11) | produces real values on a Fisher's F-distribution. (class template)
student_t_distribution (C++11) | produces real values on a Student's t-distribution. (class template)
discrete_distribution (C++11) | produces random integers on a discrete distribution. (class template)
piecewise_constant_distribution (C++11) | produces real values distributed on constant subintervals. (class template)
piecewise_linear_distribution (C++11) | produces real values distributed on defined subintervals. (class template)

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