It would be great if you could make precise what you mean by certain terms. What is a ‘closed system’ in this context? What is the definition of ‘Universe’?
What is a ‘closed system’ in this context? What is the definition of ‘Universe’?
A typical system studied in introductory probability theory might be a die roll. This is a system with a sample space with 6 states. It is an abstract thing, that doesn’t interact with anything else. In reality, when you roll a die, that die is part of the real world. The same world that also contains you, the earth, etc. That is what I meant by ‘universe’.
a closed system is a physical system which doesn’t exchange any matter with its surroundings, and isn’t subject to any force whose source is external to the system.
For a die roll, the ‘Universe’ is simply {1,2,3,4,5,6}. At least, this is the normal way it is done in probability theory.
No, I’m reasonably confident that in the example the author meant ‘Universe’ to mean {all possible states of the universe after the dice role}. The random variable X is a function mapping from states of the universe to values at the top of the die. The inverse-image#Inverseimage) of {4} is the set of all states of the universe where the die landed on 4. That inverse-image defines an [event](http://en.wikipedia.org/wiki/Event(probability_theory)). The measure of that event is what we mean when we say ‘probability of the die landing on 4’.
And it still doesn’t answer what he means by ‘closed system’.
I’m not entirely sure either. Rather than take a guess, I’ll let twanvl speak for him/herself.
I was not able to follow any of your discussion.
It would be great if you could make precise what you mean by certain terms. What is a ‘closed system’ in this context? What is the definition of ‘Universe’?
A typical system studied in introductory probability theory might be a die roll. This is a system with a sample space with 6 states. It is an abstract thing, that doesn’t interact with anything else. In reality, when you roll a die, that die is part of the real world. The same world that also contains you, the earth, etc. That is what I meant by ‘universe’.
For closed systems, I was thinking of the term as used in physics:
Is this original research or have these ideas actually been fleshed out formally somewhere I can read about them?
I don’t know. That is to say: it is original research, but probably subconsciously inspired by and stolen from many other sources.
In “Universe → {1..6}”, Universe is the type of the sample space of a random variable.
in “The value of X in a particular universe u is then X(u)”, universe refers to a specific sample in the sample space.
I don’t think this is exactly what twanvl means, especially if you consider that he’s saying things like:
For a die roll, the ‘Universe’ is simply {1,2,3,4,5,6}. At least, this is the normal way it is done in probability theory.
And it still doesn’t answer what he means by ‘closed system’.
No, I’m reasonably confident that in the example the author meant ‘Universe’ to mean {all possible states of the universe after the dice role}. The random variable X is a function mapping from states of the universe to values at the top of the die. The inverse-image#Inverseimage) of {4} is the set of all states of the universe where the die landed on 4. That inverse-image defines an [event](http://en.wikipedia.org/wiki/Event(probability_theory)). The measure of that event is what we mean when we say ‘probability of the die landing on 4’.
I’m not entirely sure either. Rather than take a guess, I’ll let twanvl speak for him/herself.