Timeline for If all the marginal distributions are continuous, then the joint distribution is continuous?
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Oct 6, 2020 at 21:00 | history | tweeted | twitter.com/StackStats/status/1313584857172373507 | ||
| Oct 6, 2020 at 15:07 | vote | accept | Star | ||
| Oct 6, 2020 at 9:38 | comment | added | John Coleman | The Wikipedia definition seems incorrect. A mixed distribution will also have uncountable support. | |
| Oct 6, 2020 at 3:57 | history | became hot network question | |||
| Oct 5, 2020 at 21:34 | comment | added | whuber♦ | @Henry That's correct. Indeed, the real numbers are usually constructed from the rational numbers as a closure process, so by definition the rationals are dense in the reals. These fundamental topological distinctions explain why characterizations of discrete random variables in terms of the cardinalities of their support are doomed to fail: the distinction between countable and uncountable supports just doesn't get to the heart of the matter. | |
| Oct 5, 2020 at 21:32 | comment | added | Henry | @whuber: Thank you. I think what you are saying is that the closure of the rational numbers is the real numbers (not something I had thought about) and that it is possible to have a discrete distribution on the rational numbers (something I had previously thought about) | |
| Oct 5, 2020 at 21:28 | comment | added | whuber♦ | What is truly interesting in the current setting is that among the discrete bivariate distributions whose marginals are supported on the entire line, there are some supported on the entire plane and others whose support is zero dimensional: that is, a collection of isolated points! (Constructing the latter is an amusing exercise.) | |
| Oct 5, 2020 at 21:28 | answer | added | kjetil b halvorsen♦ | timeline score: 12 | |
| Oct 5, 2020 at 21:27 | comment | added | whuber♦ | @Henry For intuition, see stats.stackexchange.com/a/104018/919 which describes a discrete distribution whose support is the unit interval $(0,1].$ From it you can readily construct discrete distributions supported on the entire real line; e.g., by truncating the distribution to $(0,1)$ and continuously transforming that to the line; or -- for a totally different approach -- add any distribution supported on the integers to that one. | |
| Oct 5, 2020 at 21:23 | comment | added | Henry | @whuber - I'll bite. What is a simple example of a discrete distribution whose support is the entire real line? | |
| Oct 5, 2020 at 21:15 | vote | accept | Star | ||
| Oct 5, 2020 at 23:56 | |||||
| Oct 5, 2020 at 21:13 | history | edited | Star | CC BY-SA 4.0 |
added 128 characters in body
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| Oct 5, 2020 at 20:54 | comment | added | Dave | @DilipSarwate Certainly the binary variable isn’t continuous, but the bivariate distribution has support on an uncountable set, as the Wikipedia definition discusses. | |
| Oct 5, 2020 at 20:38 | comment | added | Dilip Sarwate | @Dave A binary $Y$ can hardly be said to be continuously distributed with support $\mathbb R$ as the OP's hypotheses aver. | |
| Oct 5, 2020 at 20:36 | answer | added | Dilip Sarwate | timeline score: 16 | |
| Oct 5, 2020 at 20:20 | comment | added | whuber♦ | @Stephan Although the Wikipedia article is vague, it is evident the context of the quotation is a univariate probability distribution. The quotation is obviously incorrect in more than one dimension; take e.g. the random variable $(Z,Z)$ where $Z$ has a standard Normal distribution. Incidentally, the term "support" in the quotation is misleading. That is usually taken to be the smallest closed set having unit probability; but there exist discrete distributions whose support is the entire real line. | |
| Oct 5, 2020 at 20:14 | comment | added | Dave | @StephanKolassa What about $(X,Y)$ for normal $X$ and binary $Y?$ | |
| Oct 5, 2020 at 20:06 | comment | added | Stephan Kolassa | Wikipedia says that "A continuous probability distribution is a probability distribution whose support is an uncountable set". Under this definition, the answer is trivially yes. Are you working with some other definition of a continuous distribution? | |
| Oct 5, 2020 at 19:53 | history | asked | Star | CC BY-SA 4.0 |