Random variable and elementary events

Hello,
A sample space is the set of all possible elementary events. A random "variable" is really a real-valued function that associates a single real number to every elementary events. For example, in the case of a fair die, the sample space is ##\Omega={1,2,3,4,5,6}##. Each number is an elementary event.
A composite event would be, for example, the outcome being larger than 3: composite event {4,5,6}. Couldn't a random variable also apply to composite events like this? Or does it only assign numbers to elementary events?

When we launch two fair dies, the space of the events is

In this case, each pair is an elementary event and the numbers represent the number on the face of each die. A random variable could be defined in different ways, for example X could be the sum of the numbers on the two faces. Or the product, etc.

The sum of two numbers or their product, etc. are not elementary events in themselves but a concept applied to the elementary event. The elementary event is (1;1). The random variable is "their sum", "their product", etc., correct?

Thank you!

fog37 said:

TL;DR Summary: Trying to understand why a random variables assigns a unique value only to elementary events and not to composite events

Hello,
A sample space is the set of all possible elementary events. A random "variable" is really a real-valued function that associates a single real number to every elementary events. For example, in the case of a fair die, the sample space is ##\Omega={1,2,3,4,5,6}##. Each number is an elementary event.

Are you making these definitions up on your own? They are very limited and incorrect.
From Wikipedia: Random Variable:
"In the formal mathematical language of measure theory, a random variable is defined as a measurable function from a probability measure space (called the sample space) to a measurable space."

fog37 said:

A composite event would be, for example, the outcome being larger than 3: composite event {4,5,6}. Couldn't a random variable also apply to composite events like this? Or does it only assign numbers to elementary events?

This is not an issue for the correct definition of a random variable. Your definition is too limited and misleading in this case.

In the context of your question, part of the definition of a probability space is a ##\sigma## algebra which defines all measurable subsets of the sample space. For finite sample spaces, like a roll of a dice, this usually is the power set of the sample space. So it would include each of the individual numbers, but also subsets such as odd numbers or numbers greater than 4.

fog37 said:

A sample space is the set of all possible elementary events. A random "variable" is really a real-valued function that associates a single real number to every elementary events. For example, in the case of a fair die, the sample space is ##\Omega={1,2,3,4,5,6}##. Each number is an elementary event.
A composite event would be, for example, the outcome being larger than 3: composite event {4,5,6}. Couldn't a random variable also apply to composite events like this? Or does it only assign numbers to elementary events?

For the random variable (call it X) that you describe, each sample of X will only give you only one specific number. To deal with larger sets of numbers, you just ask for the probability that X is in {4,5,6}.

Thank you. I understand that my definition of random variable is quite basic and primitive, but hopefully pragmatic.
As mentioned, the random variable ##X## is really a function that maps an elementary event in the sample space to a (positive or negative) number. That assigned number can then be associated to another number which is its probability $$P$$. The example below shows the sample space we get from throwing two fair and independent dices. In this case, the random function ##X## is defined as "add the numbers on the faces on the two dices and assign the result to the elementary event". So event ##(1,1)## get ##X=2##.

• It is worth noticing that a random variable ##X## can be a many-to-one function: multiple elementary events can in fact be mapped to the same number (ex: ##X=4##). The numbers 2 or 4 are just two of the possible several realizations of the so defined random variable.
• Given the same sample space composed of the same elementary events/outcomes, mothing prevents us from defining the random variable differently. For example, as "multiply the numbers of the dices faces and assign the result to the event"....

On a different note, I was thinking again about a binomial experiments with a single coin where we toss the coin several times. Does each trial/toss represent a different realization of the same random variable ##X## (which has uniform probability distribution) or is each trial a realization of a different and independent random variable with uniform distribution? How do we determine this?
In general, the random variable defined in this type of experiment is "the total number of heads obtained during the ##N## tosses". This r.v. would be a binomial r.v.

Thank you!

fog37 said:

Thank you. I understand that my definition of random variable is quite basic and primitive, but hopefully pragmatic.

I don't think that it is pragmatic to invent personal definitions for standard terms that already have clear definitions.

I know what you mean. But my intro to probability textbook does not give the rigorous definition "...a random variable is defined as a measurable function from a probability measure space (called the sample space) to a measurable space."

Your example should be thought of as having more than one random variable. You can think of the selection of two numbers as taking two independent samples of a uniform random variable, U, on {1,2,3,4,5,6}. Call those samples U1 and U2. Then you have another random variable, X = U1 + U2 with the probabilities you show in post #5.
This is a very common scenario.

Well, you can always use Wikipedia before making up your own meanings:

https://en.m.wikipedia.org/wiki/Probability_space

What you call an elementary event is an outcome, which is an element of the sample space. What you call a composite event is simply part of the event space and is just called an event.

The event space includes all of your elementary events and composite events together. It has to include both since the event space must be closed under unions, intersections, and complements. Separating them as you have done breaks this important mathematical feature. For example, the complement of an elementary event is not an elementary event. So elementary events and composite events must be kept as parts of the same space.

fog37 said:

I know what you mean. But my intro to probability textbook does not give the rigorous definition "...a random variable is defined as a measurable function from a probability measure space (called the sample space) to a measurable space."

Ha! That's probably right. A simpler definition is nice. You should get used to using the simpler definition to give precise definitions for more complicated problems. Compare your original post with the description I gave in post #8: "taking two independent samples of a uniform random variable, U, on {1,2,3,4,5,6}. Call those samples U1 and U2. Then you have another random variable, X = U1 + U2"
That is routine, using standard, simple definitions. You will quickly get good at things like that and they are very common.