I am having a hard time grasping the core difference between a random variable and a stochastic process.
- A random variable assigns a number to every outcome of an experiment.
- A random process assigns a function of time to every outcome of an experiment.
But the values of this function of time can be represented with ONE SINGLE random variable as well. So what is the point in having a stochastic process when you can represent an experiment with only random variables? Could somebody make one or two examples where the difference is clear?
Appreciate it
Best Answer
Given probability space $(\Omega, \mathfrak{B}, P)$ random variable is measurable map $$X:\Omega \to \mathbb{R} $$ while random (i.e. stochastic) process is family of random variables $$X:\Omega \times T \to \mathbb{R}$$ where under $T$ often is considered as time.
On example you can understand it so: random variable represent randomness when it do not depend on time. But if it depend?