Talk:Continuous or discrete variable

a flawed and clumsy definition

begin quote:

In mathematics, variables are either continuous or discrete, depending on whether or not there are gaps between a value that the variable could take on and any other permitted values.

end quote
Really? If a variable can assume any value except 0, then that's a gap. Does that make it discrete? Michael Hardy (talk) 14:52, 30 April 2015 (UTC)

. . . and notice the two alternatives:

• "whether"
• "or not".

Which corresponds to "continuous" and which to "discrete"? If the reader thinks there's a tacit "respectively" then the first would be "continuous" since "continuous" was named before "discrete" in the opening sentence. Michael Hardy (talk) 15:12, 30 April 2015 (UTC)

The article later has "the number of permitted values is either finite or countably infinite" for discrete. That seems more rigorous. Common objection is, typical motivating examples of continuous variables are time and other physical measurements. Common reply is, we can usefully model them as uncountable, without claiming the model is literally true to actual infinity. --GodMadeTheIntegers (talk) 15:52, 30 April 2015 (UTC)

I don't really think much of this kind of article in general, and the lack of references makes it especially problematic from the standpoint of WP:OR. But if I were pressed to do so, I would probably summarize the distinction between continuous and discrete variables as follows:

In mathematics, a variable is continuous if it assumes values in a continuum (such as an interval), and discrete if it assumes values in a discrete space (such as the integers or a more general lattice).

We do not need to reinvent the wheel here. Sławomir Biały (talk) 16:03, 30 April 2015 (UTC)

Nothing wrong with this sort of article, this has to be a standard thing that we can attribute to an introductory stats textbook. I'm sure I've heard the countable/uncountable thing countless times 8*) A quick Google gives me

continuous variable: A quantitative variable is continuous if its set of possible values is uncountable. Examples include temperature, exact height, exact age (including parts of a second). In practice, one can never measure a continuous variable to infinite precision, so continuous variables are sometimes approximated by discrete variables. A random variable X is also called continuous if its set of possible values is uncountable, and the chance that it takes any particular value is zero (in symbols, if P(X = x) = 0 for every real number x). A random variable is continuous if and only if its cumulative probability distribution function is a continuous function (a function with no jumps). [1]

--GodMadeTheIntegers (talk) 16:32, 30 April 2015 (UTC)

That's a dictionary intended primarily for statistics, apparently concerning continuous and discrete random variables. We already have an article discussing that concept (namely random variable). Also, if we want to start the article "in mathematics", then the appropriate distinction is certainly not whether the variable can assume uncountable many values. There are uncountable sets that are not continua and countable sets that are not discrete. Better sources than this are presumably required. Sławomir Biały (talk) 16:46, 30 April 2015 (UTC)

This proposal from GodMadeTheIntegers won't work at all. What if the set of possible values is the set of all irrational numbers? That has plenty of gaps, so it's not continuous by the gaplessness characterization, and I don't see how one could call it discrete. Moreover, many lay readers will think "uncountable" means simply "infinite" and conclude that this article says that if the set of all possible values is the set of all integers, then the variable is continuous. A probability or statistics textbook may explain discrete and continuous probability distributions, and we already have material on that. I would characterize discrete probability distributions not by countability but be the fact that the distribution consists only of point masses, so that

$1=\sum _{x}f(x)$

where ƒ is the probability mass function. Statistics textbooks may also have other relevant material on measurement rather than on probability distributions, but that still covers only a fragment of the topic. Michael Hardy (talk) 16:52, 30 April 2015 (UTC)

Okay, fair, but fragment of what topic, though... I'd propose just changing "in mathematics" to "in statistics", though as Sławomir says we already have a better article for that, so in that case maybe just junk this. --GodMadeTheIntegers (talk) 17:27, 30 April 2015 (UTC)

Here is a possible source:[2]. On page 7, this author states that continuous variables take on values in an interval of the real numbers, while discrete mathematics involves no limiting processes (i.e. the range has no limit points) and usually involves a finite range.

I'll try to find more sources. While we may not like the vagueness, a published definition is better than our own.Brirush (talk) 17:59, 30 April 2015 (UTC)

Not a dichotomy

I recognize this is a useful intuition for people beginning math (especially applied, especially stats), but the problem is that the actual formalisms disagree: discrete is trivially continuous. Look no further than https://en.wikipedia.org/wiki/Discrete_space : we can give anything a topology and make it continuous. And this isn't actually that removed from stats: More advanced stats, e.g. working with arbitrary probability measures. uses highly related formalisms.

I don't want to derail, and especially don't want to get into a sub-field war, but at least calling out the various uses of the terminology and how they conflict would I think be useful. — Preceding unsigned comment added by Sonarpulse (talk • contribs) 05:49, 12 January 2020 (UTC)