Can one differentiate between truly random systems?
Two different random variables can have different means, different variances, different distributions. There are lots of different tests to determine whether two random variables are drawn from the same or from different distributions.
What do you mean by "truly random system"?
By "truly random" I mean that all points within a given space have an equal probability of attaining any value. The distributions you mention (like a bell curve) seem to me to be of a lesser symmetry, being describable by means or variances which themselves are not random.
Rather than a "random walk," the relation I describe might map out a truly random spatial distribution. Whether it would have nonrandom parameters of its own, I am not sure.
To be brutally honest, your concept of truly random is nonsensical and non-real. There is no such thing as a uniform distribution over an infinite range. The concept doesn't make sense.
I needed to hear that. Maybe someone will add to it.
The discrete uniform distribution, by its nature, assigns a probability of 1/n for each outcome out of n possibilities. If it were infinite, such a probability would be undefined. A continuous uniform distribution can be defined over a finite range of values, say 0 to 10, and the probability density defined as the integral b-a within this range such that b and a are real numbers and a<b. So, for example, the probability for selecting any real number between 2 and 5 would be 0.3 It's easy to see that the continuous uniform distribution cannot be infinite either for the probability of a real number between a and b to be defined.
The uniform distribution is defined by one parameter; its range (continuous) or its n value (discrete). Symmetric infinite distributions require the specification of a mean and a variance parameter. But there many other distributions used by statisticians. We might have a simple discrete distribution with just two possible outcomes such as WIN or LOSE. In a huge lottery the probability of WIN might be p=0.0000001 and LOSE p=0.9999999. Nevertheless this is still a random outcome since its not fully determined.
Any event that has a probability greater than 0 and less than 1 is random, but the uncertainty of the outcome varies according to the formula U=4(p)(1-p) so that for the lottery example, the uncertainty of the outcome for any individual participant is very low (He or she loses). Uncertainty is maximal at p=0.5.
Even over a finite set, the uniform distribution isn't always natural. For example one might estimate the probability that there will be rain tomorrow as 1/2 because there are 2 possibilities: rain or no rain. Or are there 3 possibilities--no clouds at all, clouds but no rain, and rain? Using a uniform distribution, that would change the probability of rain to 1/3, without materially changing the situation.
You give a beautiful explanation. I will try to incorporate it.
The idea of a "truly random" sequence applies to random number generators. There is no mathematical way to determine whether any sequence of numbers (or other outcomes) is "truly random". In fact no number sequence, once given, can be said to be random in a strict sense. To know if a sequence is "truly" random, you need to know how it is generated. Most computer generated sequences are "pseudo-random" in that they rely on finite algorithms.
With non-uniform distributions, including the extreme example of the lottery, some outcomes are more probable then others. They are nevertheless random because the value of any given outcome cannot be predicted with certainty. That is, each outcome is statistically independent of all previously generated outcomes. Statistical independence is a prior assumption. It cannot be proven mathematically. The best assurance of statistical independence, in my view, is based on quantum level physical processes. Quantum mechanics (QM) is based on the inherent unpredictability of individual outcomes, although they follow strict probability distributions. "True randomness" is a fundamental assumption of QM. No deviation from this expectation has been found in eighty years.
I started thread on this topic (Defining randomness? Feb 8) that goes into a little more detail.
I don't follow your argument. Statistical distributions are models. The data, and the requirements for the analysis of the data determine what model is used and how the inputs are defined. Garbage in, garbage out.
Is there a quantum algorithm or system that generates random numbers in theory?
I know about either random or non random variables. What is true random? Can you give an example of "non true" random variable other than a non random variable?
SW VandeCarr discusses this above.
Would anyone like to speculate about quantum generation of random numbers?
Quantum measurements can be used to generate truly random numbers. What else is there to say?
You're confusing a random variable with a random generator. The former is a mathematical construct defined as a function which maps a real number (defined over the interval 0,1) to a measurable space (an event space). In the discrete case p(x): x=a is the probability that x=a. In the continuous case p(x) is the probability density, usually the integral over a continuous distribution from x=0 to x=a.
The actual realization of a random process is a technical issue. The techniques used in random sampling assume that every member of a population has an equal probability of being chosen. If a population is 60% female and 40% male, a good sampling technique will produce samples which converge to this ratio. This is an indication, but not a mathematical proof of a random selection process. Random digit sequences are generated by physical processes which are assumed to be random. This usually means uniform randomness; that is, every digit has an equal probability of being generated with each unit of output. Random generators can also be made to approximate non-uniform distributions. As long as a sequence is not completely determined, it is random. The question is, what kinds of generators or sampling techniques are reliably random and how can this be measured? Mathematically, any finite sequence cannot be proven to be "truly"random.
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