What is the difference between stochastic process and random variable?

Question

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

Answer 1

A random process is a function, for instance a random walk over the plane.

A random variable has a value, for instance the average distance of the random walk after a specified number of steps.

Answer 2

A stochastic process is a family of random variables indexed by some set, usually $\mathbb{Z}^{n}$ or $\mathbb{R}^{n}$.

It's additional structure over random variables that let you establish notions of trajectories, association over a space and other interesting properties.

In empirical studies, the set may represent discrete time ($\mathbb{Z}$), continuous time ($\mathbb{R}$), geographical location in a map on a given discrete time ($\mathbb{R}^2 \times \mathbb{Z}$) and et cetera.

Answer 3

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?

Answer 4

The point is to give more information in the behaviour of your experiment. For instance if you want to describe the market value of cookies --which behaves randomly because you can't describe each actor from such market-- you could describe the law of such market value at a given time, which would certainly be some Gaussian law, and with enough effort you could even see how this law depends on the time.

BUT even if you were able to give the law of the price at each given time you wouldn't make the difference between a market value stabilizing around a fixed mean (Ornstein-Uhlenbeck process) and a dumb monkey choosing randomly at each time the price of all cookies in the world (Gaussian white noise), because your information won't describe the way the price of cookies at time $t$ influences the price at time $s>t$. It would be hidden in the coupling of your laws.

The origin of studying the stochastic processes (instead of simple random variable) comes from this need to carry the influence from the past in order to describe all the history of your experiment, and not just its value. In a sense it's the same logical step as getting from "your position is just a number (or a point in space) which I can give" to "your position is a certain function of time, thus I also get your speed, your acceleration, etc".

Answer 5

You say, "A random process assigns a function of time to every outcome of an experiment." No. A random process could be a function of time. At every time, the value is a random variable. I'll give two examples, one where the process domain is discrete, and one where it's continuous.

First, consider a gambler who repeatedly plays a game, which he wins with probability $p<\frac12$. If he wins, he gains one dollar, otherwise he loses one dollar. He continues until he runs out of money. Let $X_n$ be the gambler's bankroll after $n$ plays, where his initial bankroll is $X_0=B$ for some positive integer $B$. We can say that once the gambler runs out of money, his bankroll is always $0$, so that $X_n$ is defined for every $n\geq 0$. Each $X_n$ is a random variable. For example, $X_1$ is $B+1$ with probability $p$ and $B-1$ with probability $1-p$. The whole sequence $$X=X_0,X_1,X_2,\dots$$ is a stochastic process. (Obviously, $X$ depends on $B$ and I should really write $X(B)$ or something like that, but I suppress that.)

Here's an example of a continuous domain. Let $Y(t)$ be the temperature in a specific location on a summer day, where $t=0$ is midnight, and $t=24$ is midnight the following day. Then $Y(t)$ is a random variable, for every $)\leq t\leq24$.

I'm not sure I understand what's giving you difficulty, but you may be confusing a random variable and its value. It's true that if we take frequent readings of the temperature, we can make a graph showing what the temperature was throughout the day, but those are the values, not the random variables themselves. For example, we may find that the temperature at noon was $85^\circ$F, but that doesn't mean $Y(12)$ is the number $85$. $Y(12)$ is a random variable, distributed according to the probability distribution at that location in the summer. $Y(12)$ tells us things like, "The temperature at noon will be between $80$ and $90$ degrees with probability $.65$."