# Random Number Generator

Category: functors | Component type: concept

### Description

A Random Number Generator is a function object that can be used to generate a random sequence of integers. That is: if `f` is a Random Number Generator and `N` is a positive integer, then `f(N)` will return an integer less than `N` and greater than or equal to `0`. If `f` is called many times with the same value of `N`, it will yield a sequence of numbers that is uniformly distributed [1] in the range `[0, N)`. [2]

### Refinement of

Unary Function

### Associated types

Argument type | The type of the Random Number Generator's argument. This must be an integral type.
Result type | The type returned when the Random Number Generator is called. It must be the same as the argument type.

### Notation

`F` | A type that is a model of Random Number Generator.
`Integer` | The argument type of `F`.
`f` | Object of type `F`.
`N` | Object of type `Integer`

### Definitions

The domain of a Random Number Generator (i.e. the set of permissible values for its argument) is the set of numbers that are greater than zero and less than some maximum value.

The range of a Random Number Generator is the set of nonnegative integers that are less than the Random Number Generator's argument.

### Valid expressions

None, except for those defined by Unary Function.

### Expression semantics

Name | Expression | Precondition | Semantics | Postcondition
--- | --- | --- | --- | ---
Function call | `f(N)` | `N` is positive. | Returns a pseudo-random number of type `Integer`. [2] | The return value is less than `N`, and greater than or equal to 0.

### Complexity guarantees

### Invariants

Uniformity | In the limit as `f` is called many times with the same argument `N`, every integer in the range `[0, N)` will appear an equal number of times.

### Models

### Notes

[1] Uniform distribution means that all of the numbers in the range `[0, N)` appear with equal frequency. Or, to put it differently, the probability for obtaining any particular value is `1/N`.

[2] Random number generators are a very subtle subject: a good random number generator must satisfy many statistical properties beyond uniform distribution. See section 3.4 of Knuth for a discussion of what it means for a sequence to be random, and section 3.2 for several algorithms that may be used to write random number generators. (D. E. Knuth, The Art of Computer Programming. Volume 2: Seminumerical Algorithms, third edition. Addison-Wesley, 1998.)

### See also

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