I Random variable vs Random Process

[fog37]

TL;DR Summary

Difference between random variable and random process

Hello,

When flipping a fair coin 4 times, the two possible outcomes for each flip are either H or T with the same probability $P(H)=P(T)=0.5$.

Why are the 4 outcomes to be considered as the realizations of 4 different random variables and not as different realizations of the same random variables? A random variable is the outcome (numerical or categorical) of a random experiment. More properly, a random variable is a function that assigns a real number to each outcome of a random experiment.

I guess each coin flip is seen as a separate and independent random experiment so each outcome is therefore associated to different a different random variable...?

A random/stochastic process is generally introduced as a family of time-series where the values of the various time series at the same and specific time instant $t$ are the realization of the SAME random variables...Can a stochastic process be seen as a sequence of random variables and their realizations across different experiments? For example, we can collect the random signal temperature T(t) at 5 different locations to obtain a random process...

Could we define a sequence of coin flips a random process?

Thanks

 

[andrewkirk]

I suspect there are multiple reasons for defining it the way it usually is - ie as separate random variables.
Here's just one:
Most random processes have some form of serial dependence, so that the distribution of later outcomes is affected by earlier outcomes. If we had a framework that defined all outcomes as realisations of a single random variable, we could not write a distribution for the random variable, since that distribution would depend on which realisation we were talking about.

Summary: Difference between random variable and random process

Could we define a sequence of coin flips a random process?

We do. That is the standard definition for that case - a discrete random process in discrete time. Random processes can be discrete or continuous - meaning the outcome variable has a discrete or continuous range - and can occur in discrete or continuous time. Hence we have four different types of random process.

Summary: Difference between random variable and random process

Can a stochastic process be seen as a sequence of random variables and their realizations across different experiments?

Yes. If it's a sequence then it's in discrete time. But we also have random processes in continuous time, where there is one random variable for each instant in time, arranged as a continuum. Financiers typically model stock prices and interest rates as continuous processes in continuous time.

Summary: Difference between random variable and random process

For example, we can collect the random signal temperature T(t) at 5 different locations to obtain a random process...

No. Random processes occur across time, not across space, as the use of the word "location" implies. But a set of temperature measurements at five different times in the same place would be a continuous process in discrete time.

 

[Stephen Tashi]

No. Random processes occur across time, not across space

I'll disagree if the intended meaning of "random process" is "stochastic process". Many books on stochastic processes only deal with random variables indexed by time. However, in general, a stochastic process is an indexed collection of random variables. The index can be something besides a time stamp.

Why are the 4 outcomes to be considered as the realizations of 4 different random variables and not as different realizations of the same random variables?

The fact you say there are 4 (distinct) outcomes implies you have some way of making the distinction. So, as a matter of applied math, there are 4 different random variables. (If the first and second toss were the same random variable then the outcome of the first toss would always match the outcome of the second toss.) Sometimes people say A and B are "the same" random variable when they only mean that A and B are random variables with the same probability distribution. If you toss one coin 4 times, the first toss is not the second toss. So the two tosses are distinct random variables. If we are assuming the coin is fair then the two random variables have the same probability distribution.

A random/stochastic process is generally introduced as a family of time-series where the values of the various time series at the same and specific time instant $t$ are the realization of the SAME random variables..

No, The usual situation is that the value of some quantity at different times is considered to be a pair of distinct random variables, not a single random variable. It may be that those two random variables have the same probability distribution.

.Can a stochastic process be seen as a sequence of random variables and their realizations across different experiments? For example, we can collect the random signal temperature T(t) at 5 different locations to obtain a random process...

Yes. You can find lots of literature on spatial stochastic processes.