Random variable vs Random Process

In summary: Yes, a sequence of discrete random variables can be seen as a sequence of realizations of a single random variable.
  • #1
fog37
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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
 
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  • #2
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.

fog37 said:
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.
fog37 said:
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.

fog37 said:
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.
 
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  • #3
andrewkirk said:
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.

fog37 said:
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.
fog37 said:
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.
fog37 said:
.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.
 
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Related to Random variable vs Random Process

What is the difference between a random variable and a random process?

A random variable is a numerical quantity that can take on different values based on chance. It is used to describe the possible outcomes of a random experiment. On the other hand, a random process is a sequence of random variables that evolve over time. It is used to model the behavior of a system or phenomenon that is subject to random variations.

How are random variables and random processes related?

Random variables are the building blocks of random processes. A random process is made up of a collection of random variables that are related to each other in some way. The values of the random variables at different points in time make up the random process.

What are some examples of random variables and random processes?

Examples of random variables include the outcome of a coin toss, the number of cars that pass through a toll booth in an hour, and the height of a randomly selected person. Examples of random processes include stock prices, weather patterns, and the movement of particles in a gas.

How are random variables and random processes used in science?

Random variables and random processes are used in science to model and study complex systems and phenomena that are subject to random variations. They are used in fields such as physics, biology, economics, and engineering to make predictions and understand the behavior of these systems.

What is the role of probability in understanding random variables and random processes?

Probability is used to quantify the likelihood of different outcomes for random variables and random processes. It allows us to make predictions and analyze the behavior of these systems. Probability is also used to calculate important measures such as expected values and variances for random variables and random processes.

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