# 26 Numerics library [numerics]

## 26.6 Random number generation [rand]

### 26.6.3 Requirements [rand.req]

#### 26.6.3.6 Random number distribution requirements [rand.req.dist]

A random number distribution (commonly shortened to distribution) d of type D is a function object returning values that are distributed according to an associated mathematical probability density function p(z) or according to an associated discrete probability function .
A distribution's specification identifies its associated probability function p(z) or .
An associated probability function is typically expressed using certain externally-supplied quantities known as the parameters of the distribution.
Such distribution parameters are identified in this context by writing, for example, or , to name specific parameters, or by writing, for example, p(z|{p}) or , to denote a distribution's parameters p taken as a whole.
A class D meets the requirements of a random number distribution if the expressions shown in Table 97 are valid and have the indicated semantics, and if D and its associated types also meet all other requirements of this subclause [rand.req.dist].
In that Table and throughout this subclause,
• T is the type named by D's associated result_­type;
• P is the type named by D's associated param_­type;
• d is a value of D, and x and y are (possibly const) values of D;
• glb and lub are values of T respectively corresponding to the greatest lower bound and the least upper bound on the values potentially returned by d's operator(), as determined by the current values of d's parameters;
• p is a (possibly const) value of P;
• g, g1, and g2 are lvalues of a type meeting the requirements of a uniform random bit generator;
• os is an lvalue of the type of some class template specialization basic_­ostream<charT, traits>; and
• is is an lvalue of the type of some class template specialization basic_­istream<charT, traits>;
where charT and traits are constrained according to [strings] and [input.output].
Table 97: Random number distribution requirements [tab:rand.req.dist]
 🔗 Expression Return type Pre/post-condition Complexity 🔗 D​::​result_­type T T is an arithmetic type. compile-time 🔗 D​::​param_­type P compile-time 🔗 D() Creates a distribution whose behavior is indistinguishable from that of any other newly default-constructed distribution of type D. constant 🔗 D(p) Creates a distribution whose behavior is indistinguishable from that of a distribution newly constructed directly from the values used to construct p. same as p's construction 🔗 d.reset() void Subsequent uses of d do not depend on values produced by any engine prior to invoking reset. constant 🔗 x.param() P Returns a value p such that D(p).param() == p. no worse than the complexity of D(p) 🔗 d.param(p) void Postconditions: d.param() == p. no worse than the complexity of D(p) 🔗 d(g) T With p=d.param(), the sequence of numbers returned by successive invocations with the same object g is randomly distributed according to the associated p(z |{p}) or P(zi|{p}) function. amortized constant number of invocations of g 🔗 d(g,p) T The sequence of numbers returned by successive invocations with the same objects g and p is randomly distributed according to the associated p(z |{p}) or P(zi|{p}) function. amortized constant number of invocations of g 🔗 x.min() T Returns glb. constant 🔗 x.max() T Returns lub. constant 🔗 x == y bool This operator is an equivalence relation. Returns true if x.param() == y.param() and S1=S2, where S1 and S2 are the infinite sequences of values that would be generated, respectively, by repeated future calls to x(g1) and y(g2) whenever g1 == g2. Otherwise returns false. constant 🔗 x != y bool !(x == y). same as x == y. 🔗 os << x reference to the type of os Writes to os a textual representation for the parameters and the additional internal data of x.Postconditions: The os.fmtflags and fill character are unchanged. 🔗 is >> d reference to the type of is Restores from is the parameters and additional internal data of the lvalue d. If bad input is encountered, ensures that d is unchanged by the operation and calls is.setstate(ios_­base​::​failbit) (which may throw ios_­base​::​failure ([iostate.flags])).Preconditions: is provides a textual representation that was previously written using an os whose imbued locale and whose type's template specialization arguments charT and traits were the same as those of is.Postconditions: The is.fmtflags are unchanged.
D shall meet the Cpp17CopyConstructible (Table 31) and Cpp17CopyAssignable (Table 33) requirements.
The sequence of numbers produced by repeated invocations of d(g) shall be independent of any invocation of os << d or of any const member function of D between any of the invocations d(g).
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(g) shall produce the same sequence of numbers as would repeated invocations of x(g).
It is unspecified whether D​::​param_­type is declared as a (nested) class or via a typedef.
In this subclause [rand], declarations of D​::​param_­type are in the form of typedefs for convenience of exposition only.
P shall meet the Cpp17CopyConstructible (Table 31), Cpp17CopyAssignable (Table 33), and Cpp17EqualityComparable (Table 27) requirements.
For each of the constructors of D taking arguments corresponding to parameters of the distribution, P shall have a corresponding constructor subject to the same requirements and taking arguments identical in number, type, and default values.
Moreover, for each of the member functions of D that return values corresponding to parameters of the distribution, P shall have a corresponding member function with the identical name, type, and semantics.
P shall have a declaration of the form using distribution_type = D;