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Phrak
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What's the value of a random variable?
g_edgar said:Do you mean: "What are random variables good for" ?
Or do you mean the "value" in the sense [itex]X(\omega) [/itex] where [itex] X[/itex] is a random variable and [itex] \omega[/itex] is a sample point?
mathman said:Probability theory was given an axiomatic setting by Kolmogoroff in the 1930's. It looks in some ways like measure theory.
The experimental trial can be conceptual, like analysis of what happens after a series of coin flips by an ideal coin.
I don't know what you mean by separated.So I am curious as to the answer to the question: Can probability theory be separated from physics (in this case classical stochastic processes), so that axiomatic quantum mechanics is, in part, defined upon classical uncertainty?
mathman said:I don't know what you mean by separated.
Probability theory is a branch of mathematics. Physicists use it just like they use many other branches such as calculus, vector analysis, differential equations, etc.Phrak said:This is the pivotal question, isn't it? I hadn't been sure at the time. Now, however:-
If the axioms and theorems of probability theory are so abstract as to be free of any reference to any physical system, then probability theory is separate from physics. If there is an explicit reliance they are not.
Does this sound meaningful to you?
mathman said:Probability theory is a branch of mathematics. Physicists use it just like they use many other branches such as calculus, vector analysis, differential equations, etc.
Other fields where probability is used include statistics, finance, biology, etc.
mathman said:I guess you don't understand ordinary American jargon. Besides the point is another way of saying irrelevant.
In any case the axioms for probability theory are essentially the same as those for measure theory, with the additional condition that the total measure (probability) is one.
Phrak said:It is not irrelevent. If an axiomatic system cannot be encoded in a symbolic language, it is flawed.
Phrak said:It is not irrelevent. If an axiomatic system cannot be encoded in a symbolic language, it is flawed.
SW VandeCarr said:By encoding, do you mean the automation of the logic of probability theory (PT) to prove theorems? If so, the answer is yes. There is no flaw in PT if you accept the axioms and definitions. The definition of a random variable in particular, is that it is simply a mapping from a probability space to an event space with no guidance at all as to what value the function assigns to an event provided it is in the interval [0,1]. The following link is book sized, but the preface should at least shed some light on your question. The basic thrust is that the idea of "randomness" should be replaced by "incomplete information". There is also a good article (IMHO) on Quantum Logic and Probability Theory in the Stanford Encyclopedia of Philosophy.
http://bayes.wustl.edu/etj/prob/book.pdf
Phrak said:I'm sorry I can't reply intelligently to your post. I'll have to satisfy for myself that PT is consistent, unflawed or otherwise by putting in the effort.
SW VandeCarr said:That's an unrealistic goal. No one has proved the axioms of ZFC are self consistent or "unflawed".
A random variable is a numerical quantity that can take on different values based on the outcome of a random event. It is used to represent uncertain or random processes in mathematical and statistical models.
A discrete random variable can only take on a finite number of values, while a continuous random variable can take on any value within a certain range. For example, the number of heads obtained in 10 coin tosses is a discrete random variable, while the height of a person is a continuous random variable.
The value of a random variable is determined by the probability distribution function, which assigns probabilities to each possible value. This distribution can be derived from data, assumptions, or theoretical models.
The expected value of a random variable is the average of all possible values weighted by their probabilities. It represents the long-term average of the random variable over repeated trials.
Random variables are used in statistical analysis to model and understand uncertainty in data. They allow us to make predictions and draw conclusions based on probability and statistical methods.