Random number

What is a random sequence?
It is a sequence of independent random numbers with a specified distribution uniform distribution (equally probable).

History

• Tippett 1927 published a table of 40,000 random digits
• Kendall and Babington-Smith 1939 built a machine to generate random number to produce a table of 100,000 random digits.
• Jon von Neumann 1946 suggested the production of random number using arithmetic operations of a computer, "middle square", square a previous random number and extract the middle digits,

The sequence is not random, but it appears to be.  Sequences generated in a deterministic way are usually called pseudo-random sequences. "middle square" has proved to be a comparatively poor source of random numbers. If zero appear as a number of the sequence, it will continually
perpetuate itself.  Metropolis showed that when 20 bit numbers are being used, there are 13 different cycles into which the sequence might generate, the longest of which has a period of length 142.  On the other hand with 38-bit  numbers he obtained about 750,000 numbers that passed statistical tests
for randomness.  Many random number generators in use today are not very good.  There is
a tendency for people to avoid learning anything about such subroutines.

Super random number generator (Knuth page 4-5)
Algorithm K

This algorithm when first put onto a computer, it almost immediately converged to the 10-digit value 6065038420, which -- by extraordinary coincidence-- is transformed into itself by the algorithm (see table 1)

The moral of the story is that random numbers should not be generated with a method chosen at random.  Some theory must be used.

Generating uniform random numbers
   Un = Xn / m
Un real random numbers  uniformly distributed between zero and one.
Xn an interger between 0 and m.  Usually m is the word size of the computer.