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What's the difference between stochastic and random? I've read in the Portuguese Wikipedia that there's a difference, but I still didn't see this point on English Wikipedia.

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There is none. $ $ – Did Feb 28 '12 at 9:10
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I don't like the term "random" because its vague and people misconstrue it as "evenly distributed", but I know of no technical difference. – Alex Becker Feb 28 '12 at 9:13
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I agree with @AlexBecker. I would only add that random has many connotations (like entropy), not at all equivalent, and is a more generic term usable outside mathematics. Stochastic means nondeterministic or unpredictable. Random generally means unrecognizable, not adhering to a pattern. A random variable is also called a stochastic variable. Do random numbers exist? We speak of pseudorandom numbers. – bgins Feb 28 '12 at 9:57
    
There are random variables, random processes and stochastic processes, random differential equations and stochastic differential equations, random dynamical systems and stochastic dynamical systems etc. I never met the term stochastic variable (although you can consider a stochastic process as a random variable on an appropriate space). There may be a slight difference between random differential equations and stochastic differential equations... – Ilya Feb 28 '12 at 11:37
    
... but I think that there is no a crucial difference in the meaning, only the difference in terminology used by different groups of scientists. I can say that also in Russia the equivalent of 'random' is used in old-style literature mostly, and 'stochastic' - in the modern one. – Ilya Feb 28 '12 at 11:39

10 Answers 10

up vote 19 down vote accepted

A variable is 'random'. A process is 'stochastic'. Apart from this difference the two words are synonyms

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see here for random processes – Ilya Feb 28 '12 at 11:31
    
As if random was a subproduct of stochastic? – Voyska Feb 28 '12 at 16:58
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The term "stochastic variable" does occur sometimes. But more often "random" is used. – Michael Hardy Feb 28 '12 at 19:45

Neither word by itself has a commonly accepted formal definition in mathematics, so one cannot really ask about "the difference" between them.

They are used in phrases such as "random variable," "random walk," "stochastic process," "stochastically complete," etc, which have accepted definitions of their own. In all cases both words tend to refer to an element of chance or unpredictability. But they are generally not interchangeable; if you talk about a "stochastic walk" people will be confused.

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Stochastic comes from Ancient Greek whereas random is an old French word. (fun fact: random has totally disappeared in modern French and was replaced by aléatoire which comes from... Latin)

Otherwise there is no difference between them in the realm of Probability Theory.

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There is an anecdote about the notion of stochastic processes. They say that when Khinchin wrote his seminal paper "Correlation theory for stationary stochastic processes", this did not go well with Soviet authorities. The reason is that the notion of random process used by Khinchin contradicted dialectical materialism. In diamat, all processes in nature are characterized by deterministic development, transformation etc, so the phrase "random process" itself sounded paradoxically. Therefore, Khinchin had to change the name. After some search, he came up with the term stochastic, from στοχαστικὴ τέχνη, the Greek title of Ars conjectandi. Being popularized later by Feller and Doob, this became a standard notion in English and German literature.

Funny enough, in Russian literature the term "stochastic processes" did not live for long. The 1956 Russian translation of Doob's monograph by this name was already entitled Вероятностные процессы (probabilistic processes), and now the standard name is случайный процесс (random process).

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The terms "stochastic variable" and "random variable" both occur in the literature and are synonymous. The latter is seen more often. Similarly "stochastic process" and "random process", but the former is seen more often.

Some mathematicians seem to use "random" when they mean uniformly distributed, but probabilists and statisticians don't. I suspect those who do that haven't thought about it much.

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Any example of occurrences of "stochastic variable", WP excepted? – Did Aug 29 '12 at 15:28
    
@did : I don't have any at hand, but I've seen it in print. – Michael Hardy Aug 29 '12 at 16:28
    
@did : google.com/… – Michael Hardy Aug 29 '12 at 16:30
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scholar.google.com/… – Michael Hardy Aug 29 '12 at 16:31
    
Thanks for the links. After skimming very partly through them, what strikes me is that their majority is related to applications of mathematics (electrical engineering, management sciences, econometrics, physics, artificial intelligence, automatics, water resources research, others) rather than to mathematics and/or probability theory per se. My guess is that the frequency of "stochastic variable" would vanish, or nearly so, if the corpus was restricted to these fields. – Did Aug 29 '12 at 17:18

The term stochastic in Hydrology science refers to a process which periodically and apparently-independently happens but a kind of dependency exists. For example, if the flow of a river in last (say) 2 weeks has been low, it will probably be low in the next weeks too. So, the flow of a river is not a complete random variable but stochastic.

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Likewise, I've noticed that network theory tends to refer to traffic as being stochastic. Such traffic, from the point of view of a router, would be considered random, but of course each packet was deterministically produced. – einnocent Sep 4 '14 at 18:15

A random process is unpredictable such as the movement of the tip of a feather In wind. If we assume that the movement of a roller coaster is deterministic, then a stochastic process would be the movement of the tip of a feather attached to a moving roller coaster. That is to say, stochastic processes have components that are both deterministic AND random; e.g. Martingales.

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From a remote sensing point of view, we usually refer to a bounded but unpredictable process as stochastic. If the process were unbounded and unpredictable I would tend to use random, but this case doesn't occur very much in my world! :)

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I would make a distinction: for example in a queueing system the arrival times (or interval times) might be modelled by a Poisson process which would be time independent and would not be bound my initial conditions. This would be an example of a random process which outputs random variables. Service time in the queue would be dependent on the previous state(s) of the system and possibly initial conditions. This would be an example of a stochastic process which also outputs random variables. I am not sure that the term ‘stochastic variable’ has any real meaning except possibly to indicate how the variable was produced.

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As this is a four year old Question with an Accepted Answer (and others), Readers would benefit from your supporting any new material in your Answer with some references to the literature, etc. – hardmath Jul 23 at 1:31

There is absolutely a difference between stochastic process and randomness. For example, if I take one step then let's suppose my friend takes two steps. Now my friend's steps are not random, those are dependent on my steps. That means my friend's steps has a process which is the number of steps I take. But it is random because my friend doesn't know how many steps I will take. So the steps I take is a random walk.

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