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/
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