3 ms·

It's actually a very good question. Even up to the Renaissance, we had no real conception of "probability" -- even though there were games of chance that depen

by mturmon 11y ago

It's actually a very good question. Even up to the Renaissance, we had no real conception of "probability" -- even though there were games of chance that depended on it. The Greeks had notions of impossible/possible/likely, but no way to link the notions to the world. It was finally in the correspondence of Pascal and Fermat in 1654 that a mathematics for probability started to emerge. With the emergence of the laws of large numbers, this mathematics could be linked to observable real-world occurrence rates. (Ian Hacking's book "The Emergence of Probability" is a standard reference, http://www.jstor.org/stable/2184357?seq=1#page_scan_tab_contents. http://www.jstor.org/stable/2184357?seq=1#page_scan_tab_cont...) Suppose you were confronted with a stream of random numbers, and no intellectual or mathematical framework for quantifying them. Just coming up with a notion of a frequency curve (i.e., a density), and statistical independence of "unlinked" events would be a huge leap. I used to TA an undergrad probability class, and in the first weeks, I'd ask the students to quantify some random variable, and they'd sometimes blankly look back and say, "it's just random" -- what we taught them is how you break that down in terms of the familiar quantities (densities, independence, correlations) and operations (multiplying for independence, integrating for expectation, integrating and renormalizing for conditioning). So to get back to your original question, even a deterministic sequence, like 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, ... could be regarded as random, and given a density and all the other folderol. And a "completely random" sequence of 0/1 equiprobable bits also has a density, etc. And so does everything in between.

radarsat1 11y ago
Thank you, that is a very good answer. So "random" is how you look at a sequence. It's more a model (context in which to interpret a sequence) than an observable fact. Is that a fair interpretation? By saying "this is a random sequence," you enable a statistical view of the problem. Even for purely deterministic sequences, this can be useful. Last question, something I've never been perfectly clear on, what is the difference between "random" and "stochastic"? My understanding is that a stochastic model is a model featuring stochastic variables, which are variables that draw from some random distribution. Is the terminology I am using correct?
- mturmon 11y ago
  For almost all purposes, "stochastic" is literally a synonym for "random". But sometimes the usage favors one and not the other. You would talk of a "random variable" (loosely, a quantity that exhibits random behavior, although there is a technical definition that is more precise and has a different flavor), not a "stochastic variable". Sometimes, you would talk about a "stochastic matrix" (which is actually deterministic, an example of a namespace clash), because a "random matrix" would indeed be random be filled with "random variables". But, you would often talk about a "stochastic process" like Brownian motion. Although you can also say "random process", and that's the same thing. You would not talk about a "random distribution", that sounds like you mean "arbitrary distribution" (as in, "I picked out some random socks today"). But you don't mean arbitrary, you mean a "statistical distribution" or "probability distribution". To get all this right, you have to live in that world, you can't just visit. If an outsider could just speak the language using clear rules, there would be no point in all those advanced classes, right? ;-)
  - radarsat1 11y ago
    Thank you!

by mturmon 11y ago

radarsat1 11y ago

Thank you, that is a very good answer. So "random" is how you look at a sequence. It's more a model (context in which to interpret a sequence) than an observable fact. Is that a fair interpretation? By saying "this is a random sequence," you enable a statistical view of the problem. Even for purely deterministic sequences, this can be useful. Last question, something I've never been perfectly clear on, what is the difference between "random" and "stochastic"? My understanding is that a stochastic model is a model featuring stochastic variables, which are variables that draw from some random distribution. Is the terminology I am using correct?

mturmon 11y ago
For almost all purposes, "stochastic" is literally a synonym for "random". But sometimes the usage favors one and not the other. You would talk of a "random variable" (loosely, a quantity that exhibits random behavior, although there is a technical definition that is more precise and has a different flavor), not a "stochastic variable". Sometimes, you would talk about a "stochastic matrix" (which is actually deterministic, an example of a namespace clash), because a "random matrix" would indeed be random be filled with "random variables". But, you would often talk about a "stochastic process" like Brownian motion. Although you can also say "random process", and that's the same thing. You would not talk about a "random distribution", that sounds like you mean "arbitrary distribution" (as in, "I picked out some random socks today"). But you don't mean arbitrary, you mean a "statistical distribution" or "probability distribution". To get all this right, you have to live in that world, you can't just visit. If an outsider could just speak the language using clear rules, there would be no point in all those advanced classes, right? ;-)
- radarsat1 11y ago
  Thank you!