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The importance of random number generators has increased over the years. This follows from the fact that contemporary research methods rely more and more on simulation and the increased importance of encryption technology. The output of a random number generator is created by either an algorithm or a physical device. The most popular method for random number generation is through the use of an algorithm. This article presents a new category of physical random bit generator that is packaged by several manufacturers. A statistical analysis of the output from the generators is given.

Use of empirical studies based on computer-generated random numbers has become a common practice in the development of statistical methods, particularly when the analytical study of a statistical procedure becomes intractable. The quality of any simulation study depends heavily on the quality of the random number generators. Classical uniform random number generators have some major defects—such as the (relatively) short period length and the lack of higher-dimension uniformity. Two recent uniform pseudo-random number generators (MRG and MCG) are reviewed. They are compared with the classical generator LCG. It is shown that MRG/MCG are much better random number generators than the popular LCG. Special forms of MRG/MCG are introduced and recommended as the random number generators for the new century. A step-by-step procedure for constructing such random number generators is also provided.

The calculus of residues is one of the many beautiful tools that comes out of the field of complex variables. The calculus of residues is applied, together with the inversion formula for characteristic functions, to compute the non-gamma probability density function for the sum of gamma variates with different shape parameters. The distribution of the sum of gamma variates is needed in problems in statistical inference, as well as stochastic processes. This derivation seems more elegant than previous methods for deriving the density function of such a sum. Furthermore, the numerical computation is straightforward, especially in any symbolic computer language.

A two-stage procedure is proposed for the generation of random variates from a multinomial distribution. In the first stage we propose that cell frequencies be generated as random deviates from a Poisson distribution with parameters proportional to the probabilities of the multinomial. When the sum of the frequencies generated in the first stage is less than the total sample size of the multinomial distribution, the remaining observations are generated in a second stage from the multinomial distribution. An optimal or near-optimal set of Poisson parameters in the first stage, used in conjunction with the alias method, can reduce the amount of computation required to generate the multinomial variates to less than 10% of that required by using simple random sampling to obtain the individual observations.

The general approach to generating random variates through transformations with multiple roots is discussed. Multinomial probabilities are determined for the selection of the different roots. An application of the general result yields a new and simple technique for the generation of variates from the inverse Gaussian distribution.