What is the motivation behind random variables in probability theory?(adsbygoogle = window.adsbygoogle || []).push({});

The definition is easy to understand. Given a probability space (Ω, μ), a random variable on that space is an integrable function X:Ω→R. So essentially, it allows you to work in the concrete representation R instead of the abstract Ω.

But why is that useful?

Take a simple example of flipping a pair of coins in order. Our event set is Ω={HH, HT, TH, TT}. Our probability function is μ({HH}) = μ({HT}) = μ({TH}) = μ({TT}) = 1/4.

The random variables on this set can be a simple numeric assignment: X(HH) = 1, X(HT) = 2, X(TH) = 3, X(TT) = 4. Or it can be in a different order. Or it can be non-injective, such as the constant random variable X(s) = 1, or the "outcome same" function, X(HH) = X(TT) = 1, X(HT) = X(TH) = 0.

How is that useful for analysis? Does it have to do with the random variable's role in defining the moments of a distribution? Or perhaps random variables simply aren't useful in the finite case?

The reason I'm trying to understand this is I am (trying) to learn a little about non-commutative probability theory (and ultimately, a little about quantum mechanics).

The source I'm working off of is a paper by Mitchener (http://www.uni-math.gwdg.de/mitch/free.pdf [Broken]) which has a very succinct introduction, but after chapter 2, becomes much more abstract than I care to deal with. I'm looking for simple applications that can be modeled in software, not the hardcore theory.

Ideally, I think I'd be satisfied if I could find a concrete non-commutative probability model for the game found in Sigfpe's blog on negative probabilities (http://blog.sigfpe.com/2008/04/negative-probabilities.html).

But the first step is to understand why random variables are so important in probability theory. since non-commutative probability is described exclusively in terms of them.

Any help would be greatly appreciated.

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# Motivation behind random variables?

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