Somewhat like Computer Science, Probability Theory has the nasty habit of renaming well-known concepts from Functional Analysis to make them conform with its somewhat peculiar world view. However, Probability isn't nearly as bad; usually only ONE term is invented, and there is an innate order in the new names. In the hope of aiding the student of either field who is moving to the other (and especially to help myself remember translations I already have), I present a dictionary of equivalent terms.
- sample space == σ algebra
- The sample space is just the set of subsets of our space Ω for which P(.) is defined. As such, it's just a σ algebra.
- probability == finite measure
- Yup, it's (almost) true. A probability is just a measure P on Ω for which P(Ω)=1. If we have any finite measure μ on Ω, we can normalize it by defining P(A)=μ(A)/μ(Ω), getting a probability measure. So there's really no difference.
If μ(Ω)=∞, we cannot normalise. In particular, we cannot take Lebesgue measure on the real line and get a uniform distribution on the real numbers; the two envelope paradox shows no such creature exists.
- event == measurable set
- An event is something for which the measure P is defined; that's known as a measurable set.
- random variable == measurable function
- expectation == integral
- It's easy:
E(X) = ∫ΩX(ω) dP(ω)
- /msg ariels
- to add your favourite example (or ask about a term which you know on one side of the fence)
Some things are almost the same on both sides: Analysis often deals with L2 norms, while probability prefers to normalise the random variable first by subtracting its expectation, giving the standard deviation. It's still essentially the same thing (and the Cauchy-Schwarz inequality holds on both sides).
Probability is sometimes viewed as a branch of Functional Analysis, but that's not quite right either. At the very least, it brings in tools like the martingale. The only martingales showing up in analysis are precisely those of probability, with precisely the same name. In a broader view, the concepts of probability are... just different... from those of analysis.
This dictionary might help people see the other side.
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