We offer a procedure for generating a gamma variate as the cube of a suitably scaled normal variate. It is fast and simple, assuming one has a fast way to generate normal variables. In brief: generate a normal variate x and a uniform variate U until In ( U )<0.5 x 2 + d - dv + d ln( v ), then return dv . Here, the gamma parameter is α ≥ 1, and v = (1 + x / *** with d = α - 1/3. The efficiency is high, exceeding 0.951, 0.981, 0.992, 0.996 at α = 1,2,4,8. The procedure can be made to run faster by means of a simple squeeze that avoids the two logarithms most of the time; return dv if U < 1-0.0331 x 4 . We give a short C program for any α ≥ 1, and show how to boost an α <1 into an α > 1. The gamma procedure is particularly fst for C implementation if the normal variate is generated in-line, via the #define feature. We include such an inline version, based on our ziggurat method. With it, and an inline uniform generator, gamma variates can be produced in 400MHz CPUs at better than 1.3 million per second, with the parameter α changing from call to call.

We suggest an interesting and fast method for generating normal, exponential, t, von Mises, and certain other important random variables used in Monte Carlo studies. The right half of a symmetric density is cut into pieces, then, using simple area-preserving transformations, reassembled into a rectangle from which the x-coordinate—or a linear function of the x-coordinate—of a random point provides the required variate. To illustrate the speed and simplicity of the Monty Python method, we provide a small C program, self-contained, for rapid generation of normal (Gaussian) variables. It is self-contained in the sense that required uniform variates are generated in-line, as pairs of 16-bit integers by means of the remarkable new multiply-with-carry method.

A recently proposed test for uniform random number generators is based on the mean and variance of the outcome of a sequence of iterations. This note points out that many nonuniform random number generators would pass such a test, and derives an improved test based on the exact distribution of the outcome.

This paper discusses the testing of methods for generating uniform mlml)crs in a computer--the commonly used multiplicative and mixed congruential generators as well ks two new methods.Tests proposed here are more stringent than those usually applied, because ~he usual tests for randomness have passed severa:l of the commonly used procedures which subsequently gave poor results in actual Monte Carlo calculations.The princi-pal difl!eulty seems to be that certain simple functions of n-tuples of uniform random mlmbers do not have the distribution that probability theory predicts.TwO alternative generating methods are described, one of them using a table of uniform numbers, the other one combining two congruential generators.Both of these methods passed the tests, whereas the conventional multiplieative and mixed congruential methods did not.

This note points out the superfluity of a method of Stein and Storer for constructing a multivariate normal sample, and suggests a simple alternative.