Random number

use 
1. simulation
2. sampling
3. numerical analysis
4. computer programming
5. decision making  randomness is an essential part of optimal
strategies in the theory of games
6. recreation

sequence of independent random numbers with a specified distribution
uniform distribution (equally probable)
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, example, generate 10-digit
numbers, previous was 5772156649, square 
	33317792380594909201
the next number is  7923805949

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.