Random Number

A random number is a number chosen as if by chance from some specified distribution such that selection of a large set of these numbers reproduces the underlying distribution. Almost always, such numbers are also required to be independent, so that there are no correlations between successive numbers. Computer-generated random numbers are sometimes called pseudorandom numbers, while the term "random" is reserved for the output of unpredictable physical processes. When used without qualification, the word "random" usually means "random with a uniform distribution." Other distributions are of course possible. For example, the Box-Muller transformation allows pairs of uniform random numbers to be transformed to corresponding random numbers having a two-dimensional normal distribution.

It is impossible to produce an arbitrarily long string of random digits and prove it is random. Strangely, it is also very difficult for humans to produce a string of random digits, and computer programs can be written which, on average, actually predict some of the digits humans will write down based on previous ones.

There are a number of common methods used for generating pseudorandom numbers, the simplest of which is the linear congruence method. Another simple and elegant method is elementary cellular automaton rule 30, whose central column is given by 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, ... (OEIS A051023), and which provides the random number generator used for large integers in the Wolfram Language. Most random number generators require specification of an initial number used as the starting point, which is known as a "seed." The goodness of random numbers generated by a given algorithm can be analyzed by examining its noise sphere.

When generating random numbers over some specified boundary, it is often necessary to normalize the distributions so that each differential area is equally populated. For example, picking  and  from uniform distributions does not give a uniform distribution for sphere point picking.

In order to generate a power-law distribution  from a uniform distribution , write  for . Then normalization gives

(1)

so

(2)

Let  be a uniformly distributed variate on . Then

			(3)
			(4)
			(5)
			(6)

and the variate given by

			(7)
			(8)

is distributed as .

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