Random Number Generators

Abstract

what are called Monte Carlo calculations. Such calculations depend on having available sequences of numbers which appear to be drawn at random from particular probability distributions. For convenience we will refer to any such numbers simply as random numbers.Our purpose is to survey the problem of obtaining these sequences of numbers, with particular emphasis on the procedures used for their generation on storedprogram computers. The term "pseudo-random" is often used to describe the random numbers which are obtained on computers.We begin in section 2 with a brief indication of what types of calculations require such a supply of random numbers.Then in section 3 we turn to the main topic, which is a thorough treatment of the number theoretic properties of the methods of generation called "mixed congruential", followed for comparison by a brief treatment of the older "multiplicative congruential" methods. We find that the former have several theoretical advantages over the latter. We also refer briefly to some recent theoretical results concerning the serial correlation of the generated sequences.In section 4 we consider some of the statistical properties which must also be required of these sequences. Here the mixed methods lose some of their attractiveness. Under certain circumstances they can produce sequences which fail to pass the required tests. On the other hand the multiplicative methods produce sequences with consistently good statistical properties.In the final three sections we summarize other aspects of the subject. In section 5 we consider the problem of using numbers from the uniform distribution to obtain numbers from various other distributions. In section 6 we draw attention to several problems which seem to warrant further study. Finally, in section 7, we describe some of the historical development of the subject, and here we refer to other methods, and to other approaches to the problem.The bibliography is intended to be complete with respect to references concerned with the generation of random numbers on computers. It contains substantially all such references to the open literature, as well as many references to government and company reports. In addition it contains a number of references concerning each of the related topics which are considered in this paper. Generally speaking, Monte Carlo methods have not been particularly successful when applied to these less natural situations. However, they are indispensable in most of thoe natural applications, where there is often no alternative procedure. CALCULATIONS REQUIRING RANDOM NUMBERSA characteristic of the Monte Carlo method is that the required solution is approached with an error which is 0(n112), where n is the number of trials,

History of uniform random number generation

“…The conditions for this special case were proved earlier by Greenberger (1961). For c = 0, Hull and Dobell (1962) give an expression for the maximal possible period for any integer m > 0 and they give conditions on a and x 0 for this maximal period to be reached. These conditions follow from standard results in number theory known at the time (Ore 1948); they were already proved by Gauss in 1801.…”

“…Hull and Dobell (1962) gave maximal-period conditions for general m, a, and c > 0, and proved their sufficiency. The conditions are: (i) c is relative prime to m, (ii) every prime divisor of m is also a prime divisor of a − 1, and (iii) if m is a multiple of 4 then a − 1 is also a multiple of 4.…”

“…References to other early test proposals from around 1940 can be found in Hull and Dobell (1962). Several more have been proposed afterward, many of them designed to be representative of specific applications or to detect known structures present in certain types of generators.…”

Monte Carlo Analysis in Academic Research

“…There was quite a bit of trial and error as various randomization schemes were tried. An early review essay on computer PRNGs included 142 citations with a seemingly endless collection of proposed generators (Hull & Dobell, 1962)! A pseudo random number generator aims to select values in an "equally likely" fashion from a set of integers, usually the range from 0 to the largest possible integer that the system can hold.…”

Particle-transport simulation with the Monte Carlo method

“…Given random variables X,Y with means x, Y,respectively, then the covariance of X and Y is defined as In the method of antithetic variates, given the estimator twith expectation 8, we seek a second estimator t* having the same expectation as t but possessing a strong negative correlation with t. Then (t + t*)/2 will be an unbiased estimator of 8 …”