# Why do stochastic processes involve time? [duplicate]

We define random variables as functions on a sample space $$X(ω), ω ∈ Ω$$. Here I do not see time being involved.

A stochastic process is a family of random variables, but they also are functions of two variables $$X(ω, t)$$ where $$ω ∈ Ω$$ and $$t$$ denotes time.

When I see a plot of a random process, I notice that the family of random variables is used as X-axis (or is it just only one random variable in diferent $$t$$ times?). The Y-axis is the range of this variables assuming they all are $$i.i.d.$$

I struggle in having a moderately clear concept of what a stochastic process is!

More abstractly, a stochastic process is just a random variable $$X$$ taking values in a function space, for example the space of continuous functions with input $$[0,T]$$ taking values in $$\mathbb R$$, we typically denote this space as $$C([0,T])$$ and call it Wiener space. As a concrete example, the function $$t \mapsto t^2$$ on $$[0,T]$$ is a member of this space, but obviously not very random. So in this sense of the definition, for each $$\omega \in \Omega$$ we actually get a full trajectory, not just a value at a particular timepoint - the random variable is defined for all timepoints simultaneously.
When we think about a stochastic process at a given timepoint, we are actually thinking about the projection from the random variable down to the space it is taking values in. Concretely, we define $$t \mapsto X_t(\omega)$$ as a map taking values in $$\mathbb R$$. This is also a random variable with some associated probability distribution. One example is Brownian motion, which at every $$t$$ has a normal distribution with mean 0 and variance $$t$$, so $$X_t \sim \mathcal N(0, t)$$.