Timeline for Why do we need random variables?
Current License: CC BY-SA 3.0
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Feb 20, 2017 at 13:56 | comment | added | user21820 | @TomFitzhenry: (b) In fact I had asked about a stochastic process last year in this Math SE post that is not trivial to construct. I don't know of a good reference for what I'm describing though, because much of it is based on my own mathematical experience and background in logic. | |
| Feb 20, 2017 at 13:53 | comment | added | user21820 | @TomFitzhenry: (b) I recommend teaching probability after laying down precise axiomatization, which is a non-trivial task! One could think of a random variable $X$ as a program that returns an object sampled from some distribution, and then expressions such as "$P( X < 0 )$" and "$E(X)$" can be given specific meaning axiomatically. We also need to be careful with events; one way would be to to extend the logic so that we can reason "In any outcome, if $X < 0$ then $X < 1$." and from that conclude "$P(X < 0) < P(X < 1)$.". This axiomatization is intuitive but not trivial to justify. | |
| Feb 20, 2017 at 12:57 | comment | added | user21820 | @TomFitzhenry: The main issue I wished to bring attention to is that (as this question shows) few people actually think carefully about what properties exactly they want their mathematical objects to have. It is especially useful in pedagogy because students can learn precisely what they are allowed to do. All too often the rules are not clear to them so they just guess and try mimicking the teacher without proper understanding. | |
| Feb 20, 2017 at 12:46 | comment | added | user21820 | @TomFitzhenry: (a) In any case, even programming languages with strong type systems rarely support reasoning about correctness, so arguably they do not provide a very good role model, but it is a start; same way Hilbert's attempted axiomatization of Euclidean geometry is useful in making clear just what points and lines are supposed to be like (Note my usage of "like" is akin to the way different models of an axiomatization share some properties but not others.) My view is that the mathematical notion of axiomatization is the right way to think of interfaces! | |
| Feb 20, 2017 at 12:35 | comment | added | user21820 | @TomFitzhenry: (a) There is not a single correspondence between programming-style abstraction and mathematical abstraction, mainly because programming is far more non-formalized than mathematics. In most programming languages it is the programmer's responsibility to make sure their programs function correctly; behavioural guarantees are not even specifiable in most languages, not to say checkable. There are exceptions but rarely used in industrial applications. See softwareengineering.stackexchange.com/a/279362 for some details on the relatively powerful end of the spectrum. | |
| Feb 20, 2017 at 9:53 | comment | added | user105106 | Useful perspective. Do you have any resources on either: (a) The relationship between programming-style abstraction and abstraction in maths? (a) Probability taught random-variable-first, rather than probability-space-first? | |
| Sep 24, 2016 at 17:21 | history | answered | user21820 | CC BY-SA 3.0 |