Random and Pseudorandom Numbers

Date: 10/13/98 at 20:01:37
From: Patrick Sagebiel
Subject: Random numbers

What are examples of random/pseudorandom numbers, and what is the 
difference? When are the numbers of a sequence truly random numbers 
and not pseudorandom? What shortcomings have been observed in 
simulations and experiments in which pseudorandom numbers have been 
used? What are the properties that pseudorandom numbers can have that 
random numbers should not have?

Date: 10/14/98 at 11:29:55
From: Doctor Rob
Subject: Re: Random numbers

It is hard to define a "random number," or even a "sequence of random
numbers." It is easier to define a random process for generating a
sequence of numbers. It is a process such that the generating of the
next number in the sequence is independent of the generation of all 
the earlier or later numbers in the sequence, according to any set of
statistical tests you want to specify. In other words, the a priori
probability of generating any specific number in the range of the 
process is independent of all other numbers generated before or after, 
and distributed according to some specified probability distribution.  
If a random process is being used, that implies that the next number 
to be generated cannot be predicted more accurately using the earlier 
numbers generated than it can without them.

A pseudorandom process approximates a random one, but is not a random
one in that the number generated at some step n is dependent on the
numbers generated at steps 0, 1, 2, ..., n-1, but in such a way that
the dependence is very difficult to detect using common statistical
tests. The next number can be predicted more accurately than at random
(in fact, perfectly), but not using the common statistical tests.

This should answer your first, second, and fourth questions. As to the
undesirable properties of pseudorandom numbers observed in practice, 
the only example which comes to mind is the linear congruential 
pseudorandom number generator. It is of the form:

     x(0) = seed
   x(n+1) = A*x(n) + B (mod C)
     y(n) = x(n)/C

The pseudorandom sequence y(n) of real numbers in the interval (0,1) 
has many good statistical properties for suitably chosen A, B, and C, 
but it turns out that the pairs (y[n],y[n+1]) when plotted in the unit 
square fall into very nonrandom patterns.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/