Stochastic Process

It's helpful to define a stochastic process via an analogy to RVs. An RV $$\mathbf x$$ is a rule for assigning to every outcome $$\zeta$$ a number $$\mathbf x(\zeta)$$. A stochastic process is a rule for assigning to every outcome $$\zeta$$ a function $$\mathbf x(t, \zeta)$$. Thus a stochastic process is a family of time functions depending on the parameter $$\zeta$$ or, equivalently, a function of $$t$$ and $$\zeta$$. The domain of $$\zeta$$ is the set of all experimental outcomes and the domain of $$t$$ is a set $$R$$ of real numbers. (Note: It's not the set of real numbers, it's a set of real numbers.