Terminology: is it random?

Question

The topic of research of my master thesis is the use of probabilistic methods and models in music composition, particularly in the field of algorithmic music. As often is the case, artists tend to be quite careless when "borrowing" terms from other areas, particularly from the exact sciences, and therefore things can get confusing very quickly; moreover, the borrowed terms can get new meanings of their own in the new field (and become part of its specific terminology), so terms like stochastic music, aleatoric music, indeterminate music, chance music, etc., each imply specific types of technical or aesthetic approaches to music.

Because of that, together with the fact that "random music" is not a term used to refer to any specific artist or approach, I tend to use the term "randomness" when writing about all these approaches as a whole without singling out any of them (that is, trying to be as general as possible), meaning "music composed using indeterminate elements" which would encompass all those already mentioned approaches.

The problem is that I have seen the term "randomness" (and the adjective "random") have different meanings in a mathematical context, and that may lead to even more confusion, and this is what I am trying to clarify with this question. The meanings are:

---

1) randomness as a synonym to indeterminate. This means that any probabilistic model can be said to be random, regardless of the probability distribution behind it. That is, it does not require an uniform distribution. Some examples of this type of usage:

“In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.” My emphasis. From: https://en.wikipedia.org/wiki/Probability_distribution

Wikipedia defines a random variable as “[...] a variable whose value is subject to variations due to chance (i.e. randomness, in a mathematical sense).” See: https://en.wikipedia.org/wiki/Random_variable

“[...] the normal distribution, also called Gaussian or "bell curve", the most important continuous random distribution.” My emphasis. From: https://en.wikipedia.org/wiki/Probability_distribution

---

2) randomness as the result of a uniform distribution. Using this definition, the outcomes of a certain process is said to be random iff all outcomes have equal chance of being selected (e.g. a fair coin, a fair dice, etc.). Some examples of this type of usage:

“In situations where a population consists of items that are distinguishable, a random selection mechanism requires equal probabilities for any item to be chosen. That is, if the selection process is such that each member of a population, of say research subjects, has the same probability of being chosen then we can say the selection process is random.” See: https://en.wikipedia.org/wiki/Randomness

“A random sample is a sample in which each individual or object in the population has an equal chance of being selected.” From: http://www.mathgoodies.com/glossary/term.asp?term=random%20sample

“In statistics a sample of a population is said to be random if each member in the population has an equal chance of being chosen.” From: http://www.cut-the-knot.org/WhatIs/WhatIsRandom.shtml

---

So which is which? Is there a rigorous definition of these terms? Is using the term "random" in relation as the 1st definition above acceptable? Are there any guidelines of when these are acceptable?

Summarizing the problem, I quote this website:

In the paper quoted earlier, Kaplan et al used the following definition of random:

“We call a phenomenon random if individual outcomes are uncertain, but there is nonetheless a regular distribution of outcomes in a large number of repetitions.” From Moore (2007) The Basic Practice of Statistics.

Now to me, that does not insist that each outcome be equally likely, which matches with my idea of randomness. In my mind, random implies chance, but not equal likelihood. When creating simulation models we would generate random variates following all sorts of distributions. The outcomes would be far from even, but in the long run they would display a distribution similar to the one being modelled.

Yet the dictionaries, and the later parts of the Kaplan paper insist that randomness requires equal opportunity to be chosen.

Answer 1

It never hurts to write "uniform" or "equal probability" to make things clear.

Where there is some symmetry or no natural non-uniform distribution, it is often implicit that the probability measure should be uniform or should respect that symmetry. Unless stated otherwise, a "random" pair of distinct elements of a finite set means one chosen from the uniform measure on pairs, and a random point on the circle is chosen in a rotation-invariant way, although there are other logical possibilities for the choice of distribution.

About the Wikipedia item under (1), nobody in mathematics says "random distribution" to mean "probability distribution". Random distribution would mean selecting at random a probability distribution from some collection of distributions, according to a probability measure placed on the collection.

The examples (2) are inconsistent with the use of "random" in mathematics, and in statistics the difference between a random sample and a uniform random sample is critical. Choosing a random subsample from a known list would usually mean uniform selection and one can omit the precision by saying "random". For example in a medical experiment one might have 100 volunteers and a random 50 of them who receive the medicine. This is true uniform random selection of 50 objects from 100. But a set of telephone interviews of people with (uniform) randomly chosen phone numbers is not a uniform sample for the purposes of a sample that is meant to measure the whole population.