What's the difference between stochastic and random?

Question

What's the difference between stochastic and random?

Answer 1

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

Answer 2

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.

Answer 3

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.

Answer 4

An extreme example of a stochastic process is a deterministic signal X(t)=f(t). In this case, E{X(t)}=E{f(t)}=f(t), R(t1,t2)=E{f(t1)f(t2)}=f(t1)f(t2)

AS another example Using the Poisson points t_i, we form a process x(t) such that x(t)=1 if the number of points in the interval (0, t) is even, and x(t)=-1 if this number is odd. E{x(t)} = exp{-2landa.t}, R(t1,t2) = exp{-2landa.|t1-t2|}

This process is called semi-random telegraph signal because its value x(0)=1 at t=0 is not random. To remove this certainty, we form the product y(t)=ax(t) where a is a random variable taking the values +1 and -1 with equal probability and is independent of x(t). The process y(t) so formed is called random telegraph signal. Since E(a)=0 and E(a^2)=1, the mean of y(t) equals E{a}E{x(t)}=0 and its auto-correlation is given by E{y(t1)y(t2)} = E{a^2}E{X(t1)X(t2)} = exp{-2landa.|t1-t2|}.

Reference: Papoulis Edition 4.