# Random and pseudo-random numbers

Numbers (in particular, binary digits ) whose sequential appearance satisfies some kind of statistical regularity (see Probability theory). One distinguishes random numbers, being generated by a stochastic apparatus, and pseudo-random numbers, being constructed using arithmetical algorithms. It is usually assumed (for better or worse reason) that the sequence obtained (or constructed) has frequency properties that are "typical" for a sequence of independent realizations of some random variable with distribution function ; then one speaks of (independent) random numbers distributed according to the law . The most commonly used examples are as follows: random numbers uniformly distributed on the interval , , equi-probable random binary digits , , and random normal numbers distributed according to the normal law with mean and variance 1 (cf. Uniform distribution; Normal distribution). Random numbers with an arbitrary distribution function can be constructed from a sequence of uniformly-distributed random numbers by putting , that is, they can be found from the equation , . There are also other methods of construction: for example, it is analytically simpler to obtain normally-distributed random numbers from uniformly-distributed random numbers by using the pairs

The digits of uniformly-distributed random numbers in binary notation are equi-probable random binary digits; conversely, by grouping equi-probable random binary digits into infinite sequences one obtains uniformly-distributed random numbers.

Random and pseudo-random numbers are used in practice in the theory of games (cf. Games, theory of), in mathematical statistics, in Monte-Carlo methods (cf. Monte-Carlo method), and in cryptography, for the concrete realization of non-determined algorithms and behaviour predicted only "on the average" . For example, if the next , then a player chooses the first strategy, but if , then he/she takes the second.

It is only possible to attach a strict mathematical sense to the concept of random numbers in the framework of the algorithmic probability theory of A.N. Kolmogorov [2] and P. Martin-Löf [5]. Let be the countably-dimensional unit hypercube, let be the Lebesgue measure on and let be a largest constructively-described measurable set of measure zero (it exists). Then any sequence can be regarded as typical, and so taken as a sequence of uniformly-distributed random numbers. Similarly one can introduce the concept of constructive -typicality of an -sequence of binary symbols , , with respect to the system of all events : to be of measure not more than and of length of description not more than . It is evident from the definition that a typical sequence of uniformly-distributed random numbers cannot itself be constructive, and even the construction of an -typical sequence of random symbols requires an extraordinary large search. Therefore, in practice one uses simpler algorithms, allowing checking of their statistical "quality" with a few tests. Thus, in constructing uniformly-distributed random numbers one must necessarily test the uniform distribution of the sequence (see [3]). In simple problems, the fulfillment of certain tests can actually guarantee the usefulness of a sequence. It is sometimes more effective to use correlated random numbers, constructed from a sequence of uniformly-distributed random numbers.

Tables of random numbers and random digits have been published. However, it appears to be impossible to guarantee that they satisfy all reasonable statistical tests for non-correlation.

#### References

[1] | S.M. Ermakov, "Die Monte-Carlo Methode und verwandte Fragen" , Deutsch. Verlag Wissenschaft. (1975) (Translated from Russian) |

[2] | A.N. Kolmogorov, "On tables of random numbers" Sankhya Ser. A , 25 (1963) pp. 369–376 |

[3] | N.M. Korobov, "On some questions of uniform distribution" Izv. Akad. Nauk SSSR , 14 : 3 (1950) pp. 215–238 (In Russian) |

[4] | D. Knuth, "The art of computer programming" , 2 , Addison-Wesley (1969) |

[5] | P. Martin-Löf, "The definition of random sequences" Inform. and Control , 9 (1966) pp. 602–619 |

[6] | N.N. Chentsov, "Pseudo-random numbers for simulation of Markov chains" Zh. Vychisl. Mat. i Mat. Fiz. , 7 : 3 (1967) pp. 632–643 |

[7] | A. Shen', "Frequency approach to the definition of the notion of random sequence" Semiotika i Informatika , 18 (1982) pp. 14–42 (In Russian) |

#### Comments

#### References

[a1] | P. Bratley, B.L. Fox, L.E. Schrage, "A guide to simulation" , Springer (1987) |

**How to Cite This Entry:**

Random and pseudo-random numbers. N.N. Chentsov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Random_and_pseudo-random_numbers&oldid=14692