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